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Complex Systems - Jean-Philippe Bouchaud - Lecture 1: General Introduction. Fat tails vs thin tails — Transcript

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0:01okay
0:06okay so welcome to this uh lecture on uh
0:10statistical physics and social sciences
0:12uh obviously in very strange
0:16conditions uh so it's going to be mostly
0:19on the Blackboard but with many slides
0:23today and a few slides later on but just
0:26for illustration purposes
0:29um so this first lecture is going to be
0:32special in the sense that I'm going to
0:33spend a long time explaining what I want
0:36to do the motivation why
0:39a lecture on a link with links between
0:43statistical mechanics
0:44and economics or social sciences
0:49um
0:50and uh and and then the the lecture will
0:54be cut into Parts first part will be a
0:57lecture and the second part will be
0:58today and Valentina will intervene a
1:02little uh later on to explain
1:05how she wants to organize it
1:09um so anyway
1:11um at this point I thought it was it
1:13would be useful to explain where I'm
1:16coming from and what is my path that led
1:20me to uh
1:22study things outside physics So I myself
1:25a physicist by by training I did my PhD
1:28actually here I take a long multiplayer
1:31on physical mechanics
1:34in 85 so a long time ago
1:37and then I I slowly got interested in uh
1:43economics or Finance subjects so
1:46something that started in the 90s is now
1:49called economysics is the attempt to
1:53apply physics methods to economics and
1:57finance and in the 90s people from
2:02physics got more and more involved in
2:04the modeling of uh financial markets
2:07wealth inequalities
2:09networks agent-based models things that
2:12I'm going to talk more about during the
2:15lecture
2:17um and and so really Econo physics is is
2:21a misnomer in the sense that initially
2:23uh people were more interested in
2:25finance uh for a reason I'm going to
2:28allude to in a second rather than not
2:31economics per se and this is the evolved
2:35and and now things are changing and as
2:38more and more links between real
2:40economics and and physics
2:43so simultaneously to this academic uh
2:47foray into
2:50economics and finance
2:52I started a company uh in 94. uh called
2:58science and finance and this company
3:00merged into in 2000 with uh another
3:04company Capital fund management CFM
3:06which was launched in in 91 and is now
3:10uh a pretty large asset manager called
3:15quantitative Asset Management with
3:17around 250 people including 65 hard
3:23scientists so people with a PhD in
3:26physics mathematics computer science and
3:29so on so people maybe like you later on
3:33in your career and so this this company
3:37was really created to
3:40enhance the interaction between physics
3:43and finance but from a professional
3:45point of view this time not from uh
3:47purely academic point of view and so
3:50I've been involved in that company now
3:52since uh since 94.
3:55so why Econo physics started uh in in
4:00the 90s and not before because as you're
4:03going to see there's there's a lot of
4:06theoretical aspects that lend themselves
4:09to uh to these Bridges between
4:12disciplines
4:14but I think that what really explained
4:16the whole move was the fact that
4:18starting in the early 90s or late 80s
4:22data became available so uh whereas in
4:2785 it was really difficult to put your
4:29hand on
4:31data file for example I remember trying
4:34to get the data of the foreign exchange
4:38rates between Deutsche Mark at the time
4:41and dollar
4:42and it was pretty difficult to to get
4:45that data
4:46it was also slow to handle the data and
4:50so what started in the 90s was the
4:53simultaneous availability of enormous
4:56data sets and the possibility to do
4:59numerical experiments on on these data
5:01sets
5:03and the speed of light so to say so you
5:05wouldn't have to wait for hours to get
5:08the result of simple statistical
5:10analysis so of course this um enhances
5:13the motivation to do research when you
5:15have the tools and the data then
5:18something happens
5:20and also maybe from a more
5:24fundamental point of view the field of
5:27what's called complex systems and my
5:30lectures will be a lot of on what's now
5:33considered to be part of of the Corpus
5:35of complex System Theory which is a
5:37pretty vague notion actually but
5:41um still people started
5:44maybe in the 70s in physics to study
5:47systems that are more complex than than
5:51the usual systems that physicists were
5:53interested in before that and started
5:56devising
5:57um interesting technical tools to or
6:00technical Concepts to deal with these
6:04systems and
6:07um and I think people were convinced
6:09that these ideas could actually be
6:12exported to other fields as well and
6:15this has happened already in biology for
6:17example
6:19but um but also people really thought
6:23that something like that could be
6:25applied to economics and finance as well
6:28so uh I think that these two
6:32um items simultaneously data and
6:36theoretical Corpus pushed a lot of
6:38people in the direction of trying to
6:41look at other things rather than
6:43physical systems including man-made
6:46man-made systems like economics Finance
6:49or other sociological issues
6:53so now can I
6:57yeah okay
7:00so as I said this was uh the uh early
7:03start in the 90s
7:06but what actually happened uh was in a
7:10sense quite disappointing in in
7:13because the not much contact with
7:17mainstream economists did take place at
7:21least until quite recently and I'll try
7:24to explain why this was a difficult
7:27endeavor
7:29uh there has been some progress with uh
7:32Fringe economies so economists who
7:34themselves were not feeling its ease in
7:36their own discipline or we're studying
7:39subjects that were outside the
7:41mainstream but as I said uh this has
7:44accelerated and I'll show you a few
7:46examples of that
7:48on the other hand
7:49um
7:51not only because there's there's more
7:55data in finance or they had at the time
7:57they were there was more data to put
8:00your hands on uh in in finance there's
8:03been quite a lot of success with
8:06quantity Finance so the uh the transfer
8:09of ideas from physics to finance was
8:11actually more successful than the
8:14transfer of ideas from physics economics
8:19and in part it is because there's a lot
8:22of
8:23physicists will or or people with a PhD
8:26in physics who went into the banking
8:29industry
8:30and really had an impact in terms of
8:34practical applications of
8:36ideas or methods or concepts
8:41of the physicist that actually
8:44accelerated this interaction
8:46and so
8:48um I want to explain a little bit why
8:50it's been so difficult and in particular
8:53um insist on the different cultures and
8:56methodologies which actually surprised
9:00me early on because I thought at the
9:02time that science was a unique thing
9:04that people considered science the same
9:07way any place you would be and any
9:10discipline you you would study but
9:12actually you realize and this was my own
9:15as I said surprise that being a
9:18scientist doesn't mean that you share
9:20necessarily the same philosophy about
9:22how science should be done and this is a
9:26big huddle in the transfer of ideas and
9:30I think one has to be aware of that when
9:31one tries to uh
9:34forays in in different disciplines
9:38so let me explain
9:41the cultural gap between uh economics at
9:45least at the time 30 years ago and of
9:49course again insisting on this things
9:52are changing and changing pretty rapidly
9:54now
9:55so economics is constructed
10:00as a more of an axiomatic science in the
10:03sense that there are very strong
10:06hypothesis to start with and very strong
10:09logical constraints applied to the
10:10theory
10:11uh one reason for that and we know the
10:15same things can happen in physics as
10:16well when you lack data is that you try
10:19to supplement the lack of data by by uh
10:22by Logic by constraining the theory by
10:25imposing very strong uh logical
10:28constraints
10:29so what you end up with is a theory
10:32that's mathematically consistent
10:38and actually a lot of
10:41um theoretical papers in economics are
10:43written a little bit like math papers
10:45with a
10:46theorems and lemmas
10:49but the problem is that the the axioms
10:53or the fundamental principles on which
10:55the theory is constructed lead to ideas
10:58that are not always plausible so for
11:00example
11:01we should be rational agents with
11:04infinite foresight and infinite
11:07computing power to solve very
11:09complicated problems
11:11and as a model for human decisions
11:15um I think maybe you will share with me
11:16the idea that it's not exactly the way
11:19we humans behave anyway
11:22you can also Imagine That making these
11:25assumptions are allowed to make the
11:28theory uh mathematically tractable
11:32so the the problem with the What's
11:35called the classical construct of
11:37Economics is that there's a huge number
11:40of empirical anomalies
11:43so people speak about anomalies in
11:46finance or economics papers
11:49all the time so for example one of the
11:52best known anomaly is called the excess
11:55volatility in financial markets
11:57financial markets seem to move much to
11:59much every day compared to what should
12:03be expected in a you know kind of uh
12:05generally equilibrium rational
12:08agent point of view I can't explain why
12:12at this point but uh
12:14in a sense it's intuitively plausible
12:17that if the price of a company should
12:21reflects something fundamental about the
12:24value of this company it's very strange
12:26that this value should vary by like two
12:29percent every day on average so it moves
12:33up and down on the order of of two
12:35percent which is very large and it's
12:38usually not accompanied with some piece
12:41of news that would explain why the price
12:42is changing so
12:44so this is a well-known anomaly there's
12:46a similar anomaly for uh economies as a
12:51whole
12:51which is called the the business cycle
12:54Paradox
12:56um and it's a little bit the same that
12:58our large economies economy of the US
13:01for example
13:02is fluctuating uh far too much compared
13:05to what it what exogenous shocks that is
13:09things that should would explain why the
13:12economy goes into recessions and so on
13:15it seems that a lot of these uh crises
13:18are due to uh
13:21endogenous mechanisms rather than
13:23exogenous mechanisms at least this is a
13:26point of view that is developing right
13:28now
13:30and so all these nominees are outside
13:33the scope of textbook series including
13:35crisis and I guess that everybody will
13:37agree that understanding crisis whether
13:41they're
13:41Financial or economic economical is a is
13:47a major aspect of what the economic
13:49theory should provide
13:52uh by the way we are going to be uh or
13:56we are already now in in a crisis of a
13:58different nature but clearly the covet
14:00crisis is not uh of the type that I just
14:03described clearly there's an exogenous
14:05cause in this case
14:07but
14:08um as I'm going to show you in a second
14:12um in 2008
14:14um so maybe the 2008 crisis is already
14:18something that we all have got in view
14:20of what happened in 2020 but in 2008 uh
14:25the crisis clearly was not a very clear
14:27nature and uh and I'll show in a second
14:31uh
14:33um quotes of economists who have
14:36reflected on the fact that 2008 was from
14:38the point of view of Economics a very
14:41strange
14:42events
14:44also what is the Striking when you come
14:47from physics is that
14:49um it is very difficult to
14:51to publish uh in economics Channel if
14:54you come up with a Theory or of an idea
14:57that that is not
14:59uh you know really in the in the
15:02mainstream or or if you're not an
15:03economist yourself uh you you will find
15:06it very difficult at least until
15:08recently to publish an economics Journal
15:11so uh the discipline is the
15:15is in a sense not very open to new ideas
15:18and that's very different from uh what
15:21physics has evolved to so in order to
15:24illustrate what I just said let me quote
15:26two uh well-known economies one uh
15:30William Brita who was uh working at the
15:33bank of England for many for many years
15:35and he who wrote a little bit after the
15:38crisis in 2008
15:39so March 2009 uh a piece of which I'm
15:43extracting a few sentences here but the
15:46piece is called the unfortunate
15:48uselessness of most state-of-the-art
15:50academic monetary economics so you can
15:52imagine that the tone of the of the
15:54column
15:56and so what he says is research tended
15:58to be motivated by internal logic and
16:00aesthetic puzzles of established
16:02research programs rather than a powerful
16:04desire to understand how the economy
16:06works let alone how it works during
16:08times of stresses of stress and
16:11financial instability so the economics
16:13profession was called unprepared when
16:15this when the crisis struck I think here
16:18what is really important to underline is
16:21this sentence where he says rather than
16:24a powerful desire to understand how the
16:26economy works I think that's the
16:28difference at least what I perceived the
16:32difference between the physics approach
16:34and the economics approach is whether
16:36you really want to understand what's
16:38going on at the price of dropping
16:41a nice formalism
16:43and abandoning the idea of constructing
16:46the theory in a completely logical way
16:49or whether you prefer to have to remain
16:53in a
16:55saying mathematically well defined
16:59framework where your field advances uh
17:03through mathematics like axioms theorems
17:07and lemmas rather than
17:10ugly physics types theories so I'm going
17:12to go back to that
17:14so the other piece that I want to
17:16mention is a piece by Olivia
17:19was Chief Economist uh had the
17:24World banked I think
17:28um and he wrote something uh in 2014
17:32that you can find on the internet
17:34no IMF not a while back it's a
17:36nationally monetary fund sorry uh call
17:39where danger lacks so that's the piece
17:42in in full that you can easily access on
17:46the internet and he said something that
17:49I'm going to reiterate a little later he
17:53said we in the field field did think of
17:56the economy as roughly linear constantly
17:58subject to different shocks constantly
18:00fluctuating but naturally returning to a
18:02steady state over time so it's really
18:05this picture of a marble in a ball and
18:08you shake the ball so the marble moves
18:10but it's always attracted to the bottom
18:13of the of the ball and and therefore
18:16nothing much can happen and you'll see
18:18that this pie dying of the harmonic
18:21oscillator so to say of
18:24linear participation around a stable
18:27equilibrium is also very much a paradigm
18:31that that overwhelmed physics for many
18:33years
18:35so what he says then he says the main
18:38lesson of the 2008 crisis is that we
18:40were much closer to dark Corners
18:43situations in which the economy could
18:45badly malfunction than we thought now
18:47that we are more aware of
18:49non-linearities and the dangers they
18:51pose we should explore them further
18:53theoretically and empirically trying to
18:55create a model that describes crisis may
18:57be beyond the profession's conceptual
18:59and Technical reach at this stage so I'm
19:02really quoting these people who are well
19:06trained and well
19:08regarded economists to say to to
19:11illustrate the fact that it's not me
19:14from an excellent point of view it's
19:15always easy to um
19:18criticize other people's Garden
19:23but what I want to illustrate with these
19:26quotes is that the critics comes from
19:29inside the profession as well and it
19:32really these critics really developed
19:35since the 2008 crisis which acted as a
19:39kind of uh uh a catalyst to develop new
19:45ideas and abandoned
19:47simple theories
19:50so I've tried to explain how Economics
19:52work at least theoretical economics of
19:55course economics is a huge field with
19:57people doing uh very much empirical work
20:01and or actually field work and very far
20:04from uh Theory but uh but theory is an
20:08important aspect because people are very
20:10influenced by the general uh Concepts
20:14that are taught at an early age and so I
20:17think that even if people are not
20:19serious themselves they were still
20:22exposed to theoretical ideas that that
20:25influenced their uh their their way to
20:28do science much longer after they've
20:31been exposed to these ideas and it's the
20:33same with physics and the way most of
20:36you have learned physics I guess so
20:38physics Works a little differently I
20:40mean the observation is uh is the
20:43starting point and then there's a
20:45mathematical transcription uh of reality
20:49and from this mathematical
20:50transcriptions one tries to get
20:52predictions
20:53and then there's a loop if predictions
20:55are not compatible with observation uh
20:58kill the theory and if the predictions
21:00are compatible then carry on even when
21:03the theory is not logically consistent
21:04and this may be a surprise for people as
21:08I said people
21:09depending on their background don't
21:11regard science the same way and I guess
21:15this is strange to think that we live
21:18physicists with theories that are not
21:20logically consistent or have sometimes
21:22difficulty with the logical consistency
21:25for example
21:26the fact that thermodynamics leads to an
21:30error of time whereas is well known uh
21:34hamiltonian Dynamics is reversible this
21:37has creates an enormous havoc in
21:40statistical mechanics and it's still
21:42something that people debate how can we
21:45at the same time speak about things that
21:47are that have an arrow of time where
21:50when the microscopic foundations of the
21:53series doesn't
21:55and okay so it's it's a philosophical
21:58problem if you want but we have to learn
22:00to live with that and we're not obsessed
22:02with this uh Paradox when we use
22:05thermodynamics to actually describe
22:08what's going on in the world we have
22:11difficulty generalizing quantum
22:12mechanics from the micro scale to the
22:14macro scale we have difficulty merging
22:17gravitation with quantum mechanics and
22:19so on but as I said it's it's not a
22:22reason to stop and to not to use these
22:25uh these theories that are also
22:28extremely efficient in describing what
22:31they want to describe
22:33so this idea of efficiency of the
22:35description rather than logical
22:37consistency I think is is what is for me
22:41it at least is the one of the strongest
22:43cultural gaps between the two fields and
22:47what I've said it's very difficult to
22:49publish in the economics journal and in
22:51physics it's a little bit the other way
22:53around there's many crazy papers are
22:56accepted and published and many of these
22:59papers are actually wrong but I think
23:01that in physicists have learned that uh
23:05crazy ideas can be right and that in the
23:08process of developing science it's
23:11important to let these ideas uh at least
23:13be expressed and maybe rejected if they
23:17lead to nowhere but the idea that
23:19science progresses also by trial and
23:22error is is important we've had you know
23:25many surprises in in physics when people
23:29in the late 19th centuries thought that
23:32everything was solved and then suddenly
23:34uh new crazy apparently crazy things
23:37happen and we have to revise the theory
23:41okay so as I said there's a cultural Gap
23:44but the Gap is narrowing in 2008 uh the
23:492008 crisis has led to introspection
23:53um agent-based models ABM that's really
23:56something that physicists do all the
23:59time we come we start from micro rules
24:03so for example atoms colliding at the
24:06micro scale and and then derive or try
24:09to derive what happens at the macro
24:11scale
24:12and so agent-based models in economics
24:15but also in other disciplines take human
24:18beings as atoms but it's really the same
24:20idea how do you construct a theory of
24:23traffic jams or or epidemic spreading or
24:27maybe
24:28economies as a whole starting from
24:31Agents that do things that may not be
24:34completely rational but at least that
24:36you can simulate and try to understand
24:38sometimes with more advanced theoretical
24:42tools how simple rules at the micro
24:45level can lead to extremely complex or
24:48extremely surprising uh emergent
24:51phenomena at the macro scale
24:54um
24:55I said earlier that in the early 90s
24:59people thought that complex theory was
25:01ripened off
25:02to make a transition between physics and
25:07other fields actually we now have 30
25:09more years or 40 more years of this
25:12Theory developing
25:14with many new ideas and many new uh
25:17tools as well or at least tools that
25:20that have ripened and people understand
25:23better what they mean and so I think
25:26really
25:27um now time is right to to
25:31engage in in a constructive interaction
25:36with
25:37with economists and there are huge
25:41challenges ahead to construct something
25:43that's both intellectually satisfying
25:46and useful for policy makers
25:51and that's why I believe that we need to
25:54be cultural students students who know
25:56uh
25:57physics you know the mythology of
26:01physics but also
26:03um the mythology and the in the
26:05questions that economists ask and that's
26:09a little bit my motivation to create
26:12this course this lectures here is to
26:16have students like you who may well
26:19engage later on in these uh in these
26:22problems and and come with a cultural
26:26background that allow them to make make
26:29a difference so something I wanted to
26:31show as an illustration of this
26:34this this move or this Progressive
26:38closing of the cultural Gap
26:41is the cover page of
26:44a journal called the Oxford review of
26:46Economic Policy uh February 2018 so not
26:50long ago called rebuilding macroeconomic
26:52Theory so it's pretty ambitious and
26:54tells you again that people seem to be
26:56aware that there is a problem and if you
26:59look closely you'll see here
27:02that there's the um an article written
27:06by um the head of research at the bank
27:09of England and the aldain and a
27:11physicist called Arthur Terrell called
27:14an indiscipline interdisciplinary model
27:16of microeconomics where they Advocate
27:20the use of these agent-based models in
27:23in economics as well so just to show you
27:26that things are moving and there's a now
27:29a clear
27:33a clear view that economic theory as it
27:36was conducted before 2008 cannot
27:40continue unchanged in that we need to
27:43roll our sleeves and build something
27:46better that is able to
27:49monitor economic crisis in a more
27:52efficient way
27:55just Okay so
28:00um the scope of the lectures is
28:03to give you a series of
28:07of inspiring stories uh and of course
28:10I've said that there's no consistent
28:13framework uh
28:16yet to replace the the old
28:20rational agent way of seeing the world
28:22in economics but I think that there's
28:27enough
28:28in the Corpus of complex systems in
28:31terms of examples of these uh emerging
28:35phenomena of these widely fluctuating
28:38objects
28:39that we we can derive or build from them
28:44something that is going to be more
28:47useful and at least even if that's not
28:49the case I think these stories that I'm
28:52trying to that I'll try to give you are
28:55interesting on their own and even if you
28:58don't do anything in social sciences
29:00economics later on I hope that you'll
29:03find these stories interesting and
29:06useful for
29:07all the fields as well because actually
29:09they are pretty generic uh in terms of
29:12uh mechanisms in mathematics
29:15so
29:17um my lecture is really not about
29:21ethereums and Mathematics at all it it's
29:25not nor it is what I'm doing now
29:29something you know soft and and
29:31qualitative it's it's going to be I hope
29:34a mixture of both so I'm going to give
29:36you qualitative arguments and tell you
29:39stories but of course I'm going also to
29:42go into details sometimes on in on the
29:45board and give you real calculations of
29:47how one actually derives on things and
29:50these real calculations will be
29:52enhanced by Valentina Valentin STD where
29:56you really will have to work on
29:58uh um
30:00concrete
30:02models like like physicists are are used
30:07to so um
30:09so it is going to be quite a theoretical
30:12uh set of lectures as well
30:15um and and the basic punch line of these
30:18lectures is that if you have something
30:21that mixes fluctuations and interactions
30:24then interact interesting uh phenomena
30:28can appear
30:30so
30:32um the outline very quickly uh
30:36and how the slides will be available for
30:39you to to see
30:41um so I'm going to start today with uh
30:48discussing different types of random
30:50variables random variables with
30:53so-called thin tails and and random
30:57variables with fat tails and I'm going
30:59to tell you how they are different so my
31:02fluctuations versus y fluctuations
31:06then I'm going to move on to study
31:09multiplicative models for population
31:12growth and wealth growth or and you'll
31:14see that many of these
31:17situations are described by a kind of
31:19unified framework where many interesting
31:22mathematical uh phenomena or physical
31:26phenomena happen that I think are very
31:29useful to know about again not
31:31necessarily for social sciences but more
31:34generally in physics or other Natural
31:37Sciences
31:38I'm going to speak about uh it's called
31:41branching processes and Ava launches
31:43like that sometimes one single event can
31:47trigger
31:48an avalanche full-scale Avalanche
31:52like the word says a fraction of the
31:57slope of a mountain
31:59unpins and creates something big and of
32:03course here we're going to be interested
32:04in in the mechanism that can lead to
32:07crisis from that point of view of of one
32:10grain
32:12uh triggering more grains to to roll
32:16down the slope and so on
32:18I'll speak about uh networks in crisis
32:22how networks can lead to Contagion
32:25effects or mediate contagions
32:27and and actually again
32:30generate system-wide
32:33crisis rather than small local
32:36perturbations
32:39uh I'll speak about
32:41um interactions and stabilities and
32:43Collective effects
32:45illustrated by many different types of
32:49models and then if I have time but I
32:52never have time so um
32:54this is uh this is a promise I shouldn't
32:57even try to
32:59mention uh in my last part in the last
33:03part of these lectures I wanted to speak
33:06about I know the Dynamics of financial
33:08markets but it turns out that
33:10I don't really have time to do that
33:13anyway
33:14uh so again today is special I'm going
33:18to give a full three-hour lecture or
33:22something
33:23um with a pause uh around yeah in 10 30
33:28quarter to 11. but then after that
33:31you'll have me again
33:33um but usually I'll give a one and a
33:37half hour lecture from nine to
33:4110 30. so usually I start at 905 but now
33:45the situation is a little different and
33:47people seem to be more on time
33:50when it's online than when it's for real
33:53so I'll I'll start I'll try to start
33:55slightly after nine but not
33:58a very long after nine and then
34:02um then Valentina will do the today
34:06so from
34:0911 to uh 12 30 I guess
34:14um so maybe my lecture will be 1 hour 45
34:16and but it's it's around these uh these
34:20times and maybe I can leave her uh now
34:23to tell you a little bit what she wants
34:25to talk about and how she wants to
34:27organize these uh yesterday
34:33so Valentina is in the room with me
34:34actually but okay I hope you'll see her
34:37hear me can anybody tell me if you can
34:40hear me now
34:43I can hear you
34:46okay great so well first of all uh good
34:50morning everybody I'm Valentin and I
34:52will be working together with you on the
34:55today's and exercises
34:57that indeed we'll start next week so
35:01today I will just take two minutes to
35:03tell you something about the
35:05organization so there are four things
35:07that I want to tell you and the first
35:09one is that I think it's a good idea to
35:12have a mailing list with all the people
35:14who are here so that we can exchange
35:16emails and information about the course
35:18so I already took the email addresses
35:21which I saw which Medina Mart gave me
35:24but I think there are more people in
35:26here than the ones I have so if you know
35:28that you're not registered in that list
35:31of the people who want to follow the
35:33course please write your email either in
35:36the chat or send me an email I will
35:37write my address
35:39in the chat
35:41so that I can add you and we can be
35:45informed without going through Medina
35:48okay so that was the first thing the and
35:50I will send an email tonight maybe to
35:52check that uh everybody's there and
35:54things work
35:55so the second thing is about the
35:57material of uh the today so this year we
36:00have this repository of ens where we can
36:03put materials together with the video of
36:06the talks so I think it is a good idea
36:08to use that so we have a unique place
36:10where to store things so what I will put
36:14in there are essentially the text of the
36:17Theta so the exercises are splitted into
36:20two categories let's say we will have
36:23the today's which are normal exercises
36:25that we will discuss together on
36:27Wednesday at the Blackboard and then
36:30there are what I call the homework and
36:32the arm work are some little coding
36:36exercises which I wrote in in jupyter
36:39notebooks so in Python and they are a
36:42little bit for you to let's say play
36:45around with some of the topics that then
36:47we will discuss during the Terrace and
36:51also during the lectures
36:54um and also to visualize more
36:57simulations of the various processes
36:59that we will discuss
37:02and they are not compulsory but I think
37:04they are a good way to uh practice a
37:07little bit with these Concepts so as I
37:09said I made them in Jupiter notebooks so
37:12maybe not everybody is used to it so if
37:15you go to the folder of ens you will
37:17find a PDF with some instructions on how
37:20to download
37:22this python or these notebooks and also
37:26how to use them if you don't want to
37:27download them you can use them in the
37:29browser so there are instructions on
37:31those as well and if you don't want to
37:34use Python at all I will also upload
37:36some HTML version of the exercises that
37:39you can just read to see what's going on
37:41and then maybe you can use whatever
37:43language you want to do the exercises
37:46so for the timing it works like this I
37:48will upload the text of the today end of
37:51the arm work one week in advance so if
37:54you go there you already find the ones
37:56for next week so you have one week of
37:59time to to to read them to uh to go
38:03through sometimes in the theater there
38:04is some Theory so to go through that and
38:06try to think about the exercises then on
38:08Wednesday we discuss them and as soon as
38:10we are finished with the lecture I will
38:14upload a version with all the solutions
38:16of both today and exercises plus the
38:19text for for the ones of the following
38:21week
38:22and if you go to the folder now you will
38:25also find a little bit of outline of uh
38:28that it is for the various weeks so you
38:30can have an idea of what we are going to
38:33discuss and I also put the lecture notes
38:35of the course of Professor busho and
38:39Mark mezar at the call Polytechnic which
38:41had a little bit of overlap with what
38:43we'll be discussing here so maybe that's
38:45useful for you uh third thing I I will
38:50make a Google doc which I called
38:51question and answer for for us to to
38:55write down any comments or questions on
38:57the lecture since we kind of discuss in
39:00person so I will send that to you
39:02together with a mailing list and the
39:05last thing is that so the today is
39:07supposed to be of in theory of one hour
39:10from 11 to 12 but as you will see there
39:13is more material than what we will be
39:15covering in one hour so we can do two
39:17things either we make it longer and that
39:20is perhaps what I would prefer and and
39:23go until 12 30. if everybody agrees or
39:27we keep one hour and then with the
39:29solutions you you try to feel the gaps
39:32of what we don't manage to discuss so we
39:35can discuss about this I guess next week
39:38because I think this week everybody will
39:41follow different courses and then you
39:43make up your ideas about the schedule
39:45and so next week is perhaps a good point
39:47to decide all together
39:50okay I think that's it basically
39:54so see you next week
39:56see you yes so thank you Valentina
40:01um okay so that's the parts of the
40:04regular weeks today and lecture there's
40:08of course an exam at the end and so the
40:10exam is uh traditionally a scientific
40:13paper that recent I mean the last 10
40:17years that in English that we ask you to
40:21read comment and explain so uh there's
40:24time to read the paper and then there
40:26are various questions allowing you to
40:29express what you understood but also
40:31redo some calculations or calculations
40:34that are that are not fully explicit in
40:37the paper that we ask you to be able to
40:39redo and of course the paper is chosen
40:42uh in line with the types of models and
40:47methods that you have seen during the
40:50lectures under today
40:52so
40:54um the idea here is that usually we have
40:56a diverse crowd of students and and so
40:59some of you are more on the technical
41:02side others are more qualitative and
41:06um
41:08more on the idea science if you want and
41:11so it's it's a way not to bias these
41:16exams task one profile
41:18uh that's the other so
41:20it usually works quite well and um so
41:24yeah
41:26a few years ago we even had two um
41:30two exams I mean a choice between two
41:32exam one scientific paper to read and
41:34one traditional question and answer type
41:37of exam but this takes a lot of time and
41:40I think that we will not do that except
41:42if we're very courageous anyway
41:45um so something new compared to last
41:47years and so Valentina said that there
41:51is a a set of lecture notes that we
41:54wrote with Mark miza when we were giving
41:56a similar but quite different actually a
42:00set of lectures that I got pretty
42:02technique a few years back uh called
42:04complex systems so you can of course
42:06success that and read that but I've
42:10embarked in in actually writing
42:13lecture notes for this to take down
42:18lectures
42:20um so it's not finished far from it it's
42:22in construction but I have a few
42:24chapters
42:25that are already written and they're not
42:30final in the sense that I am still I
42:34don't have the figures there are no
42:36references and
42:38um but maybe they are going to be useful
42:40nevertheless so I'll give them to
42:43Valentina so that you can access them
42:46and of course
42:48um Sciences they're in primary form and
42:51I'm sure that your feedback will uh help
42:56improving these lecture notes for her
42:58the laser generation of students and
43:01possibly hopefully writing a real book
43:05out of them
43:07okay so that's about it for the kind of
43:10General introduction so let's now
43:13dive into uh more technical stuff just
43:18repeating to start with things that I've
43:20said
43:21in in words before uh now trying to be a
43:25little more concrete so the standard
43:28Paradigm both in in physics
43:32uh
43:33let's say most of the 20th century
43:37physics before 1960 or 70s say was built
43:41around ideas of equilibrium of gaussian
43:43fluctuations around the equilibrium
43:46and continuous Dynamics so I've shown
43:48already this marble in a bowl type of
43:52Paradigm so harmonic oscillator if you
43:56want and so in physics the usual
43:59description of that is through What's
44:01called the longer equation so the larger
44:03equation tells you that the evolution of
44:06the position of a of a particles for
44:08example a one-dimensional
44:11marble in the one-dimensional bowl if
44:14you if you want is given by two terms
44:17and we'll see that in different contexts
44:20later on
44:21What's called the false term a drift
44:23term that's a deterministic part of the
44:25evolution
44:27and in this particular case f is just
44:29minus K times x so it's a harmonic Force
44:33pulling back the particle towards the
44:36equilibrium at x equals zero but on top
44:39of this restoring Force
44:43which is very generic which just says
44:45that equilibrium is stable and so there
44:47are forces bringing back the system's
44:50house equilibrium of course this is
44:52one-dimensional but you can imagine that
44:54this is true also for higher dimensional
44:56systems but then on top of that there's
44:59usually what's called a larger noise in
45:01physics that is something random
45:03that mimics again in physics the role of
45:06temperature the role of
45:08unpredictable shocks with uh thermal
45:12molecules that agitate the system and in
45:16this Largemouth noise can have a very
45:19important consequence on the long-term
45:22evolution of the system but in this
45:24particular case in the case of a
45:26harmonica Slater nothing much can happen
45:28because when the particle moves too far
45:31because of the effect of the long run
45:33noise it's brought back Towers the
45:35origin by the harmonic force and so in
45:38the end what you find if you solve this
45:40simple linear language equation
45:44is that the position of the particle
45:46across time does like this so it
45:49fluctuates with some correlation time so
45:52spend sometimes
45:54X positive and sometimes x negative so
45:57here T the x-axis is time and the y-axis
46:00is position
46:01uh and if you do the histogram of the
46:04different positions of the particle you
46:07find a
46:08hum-shaped gaussian curve which I'm
46:12going to comment in a second but as we
46:14know the gaussian S tails that Force
46:16extremely rapidly and so okay this is
46:19the the standard Paradigm nothing much
46:21happens there are fluctuations no
46:24extreme events uh extreme events are
46:27suppressed
46:28very forcefully by the drop of the
46:30gaussian distribution and that's that's
46:33the standard Paradigm as I said again in
46:36many economics model that's the way the
46:39world is pictured there's an equilibrium
46:41and shocks that are not thermal but due
46:44to anything that you cannot describe
46:47fully so something happening maybe the
46:52covid maybe an earthquake maybe uh
46:54something else less extreme and and then
46:58evolution of the economic system as a
47:00whole is of course slightly more
47:03complicated but basically given by the
47:05same type of harmonic Force description
47:09and then the non-standard pipeline that
47:12as I said emerged in physics already
47:14some decades ago and in fact in
47:18economics as well but without having
47:21such an impact on the on on the
47:23discipline
47:25is a is a paradigm where things are out
47:28of equilibrium rather than close to
47:29equilibrium uh there are fat tails in
47:32the distribution uh
47:35trajectories have discontinuities and so
47:37here I'm showing something that will
47:40comment on later which is called the
47:42levy flight and think of that as for
47:44example a model of financial markets
47:47where
47:48the trajectory seems to be some somehow
47:51regular and then there are jumps of all
47:54sizes uh in the on the bottom graph here
47:58you see just the derivative or the
48:00numerical derivative of the
48:03of the blue line where you see these
48:05huge spikes corresponding to uh
48:08important events and
48:11um
48:11and the idea that the gaussian
48:13distribution or thin tail distribution
48:16more generally are not necessarily
48:19um
48:20the rule in that other types of
48:23distribution more much more violent can
48:27be relevant has been actually has a long
48:32history in science starting with uh like
48:35Paul Levy who was a mathematician and
48:37I'll speak about olivi's
48:40uh theorem about the central limit
48:44theorem in the case when there are fat
48:46sales I'll come back to that later
48:48Mandel brought
48:50whom I guess many of you have heard
48:53about has also pushed in that direction
48:55in particular for financial markets as
48:58early as uh 1963
49:01many others and here I want to quote
49:04famous economists at least someone
49:08I think was maybe physicists don't know
49:12enough Keynes Keynes is the analog I
49:15would say of uh people like I don't know
49:17pain man in physics I mean someone who
49:19has had a tremendous vision and on the
49:23things that you are already
49:27breathtaking in a sense and so he wrote
49:30in the 30s we are faced at every time
49:32speaking about economics and social
49:35systems well faced at every time with
49:37the problems of discreetness of
49:38discontinuity the whole is not equal to
49:41the sum of Parts small changes produce
49:44large effects the assumptions of a
49:45uniform and homogeneousb Continuum are
49:48not satisfied so you see that in these
49:51sentences you already have all the ideas
49:53that I've alluded to and all the ideas
49:56of complex systems as well oops
50:00sorry
50:01the whole is not equal to the sum of the
50:04parts this is really the idea that there
50:06are emergent phenomena that we cannot
50:08anticipate if you are only looking at
50:11particles but you there's there's a
50:14another level of description where
50:16complete new stuff happens and and this
50:19idea that
50:20um Continuum descriptions are not
50:22necessarily uh valid and my little graph
50:26here of uh of a trajectory with jumps
50:29shows you that indeed there are
50:31sometimes objects that cannot be
50:34considered as as discontinuous in the
50:37and that's going to be important so I'm
50:39now going to move to the board
50:41and then come back to the screen with
50:44more
50:45um examples
50:49of things I want to show to you soon
50:53okay
50:54so do you see the board and
50:57so I have written the general outline of
51:01last lecture
51:03introduction two types of distribution
51:05many examples generalized CLT Central
51:08limits here and concentration but at
51:10this stage I would like to you to tell
51:12me if it's too small or if it's okay
51:15otherwise I can write
51:18bigger
51:21so can you give me feedback about the
51:23size of I think a bit bigger would be it
51:26would be better even though I can read
51:28it but uh I think a slightly bigger
51:30would be more okay okay fine fine
51:33because it's very difficult to know
51:34exactly what you see so so I'll write
51:38bigger
51:39okay
51:55so I can hear you if you if you don't
51:58see done something and Valentino also
52:01tell me if something goes wrong so of
52:03course it's the first time ever I'm
52:05doing this the lecture on the board with
52:07uh nobody listening
52:09so don't hesitate to shout or to write
52:12something on the chat and then Valentina
52:14will will intervene
52:17Okay so
52:19as I've just shown you there are
52:22two two broad types of phenomena one is
52:26the kind of gauten Paradigm where
52:29um things are continuous where
52:31distributions have entailed and then
52:35this continuous phenomena with fat tails
52:37and so what the first thing I want to
52:39express more mathematically is
52:42is to describe two types of probability
52:46distributions so I'm going to write that
52:49X is a random variable
52:59so X is a continuous random variable so
53:02it could be the position of a particle
53:04it could be
53:05the wealth of an individual it could be
53:09whatever you want
53:10and I'm going to distinguish
53:14thin-tailed
53:19distribution
53:26from fat tail distribution
53:40so the random variable X will be
53:42distributed according some to some
53:45density rho of X so as usual
53:49rho of X is such that the probability to
53:52find X within a small interval X X Plus
53:56DX is given by row of X DX
54:01and fin tail distributions are
54:03essentially distribution such that all
54:06the moments of X are finite
54:10so I'm going to call
54:12MN is the nth moment of X so it's
54:15defined as the integral
54:18from minus infinity to plus infinity
54:21maybe some sometimes a variable is
54:23positive so it means that row of X is
54:26zero for x negative but in general
54:29I'm integrating over all possible values
54:32of X so the moment of X the nth moment
54:36of X is integral DX x to the N Ro of x
54:43and for this moment to be finite
54:46requires rho of x to Decay fast enough
54:49when X goes to Infinity
54:52because otherwise this integral may not
54:55be convergent and we'll see examples of
54:57that
54:59so instead of you know remaining
55:01completely General let me give examples
55:04so as I just said the most famous thin
55:08tail distribution is the gaussian so
55:11uh row gaussian of X so capital G is for
55:15gaussian so would be 1 over square root
55:19of 2 pi Sigma squared exponential of
55:23minus x minus m
55:26squared over 2 Sigma squared
55:30and of course this has a famous
55:34Bell shapes with a very thin tail
55:41you go back to the thinness of the tails
55:44in a second so the distribution is
55:46centered around its mean M and has a
55:50width
55:51Sigma and actually more technically M1
55:55the first moment of the gaussian is
55:58equal to m
55:59and and to the second moment is M
56:03squared plus Sigma squared
56:08okay but all MNS are finite
56:14because the the decay of the gaussians
56:18for large x
56:19is sufficiently fast to kill the growth
56:23of x to the N for any finite n
56:27and so okay
56:30um just to give you an illustration of
56:33uh how fast this distribution decays if
56:36you're looking at What's called the 10
56:37Sigma event so if you're looking at an
56:40event that is
56:43greater than M plus so got that I need
56:48to write bigger
56:51[Music]
56:56so if I look at an event X star which is
57:00greater than M plus 10 Sigma
57:05then the the probability to observe such
57:08an event is uh 10 to the minus 22
57:14so if you're you know 10 Sigma away that
57:17means okay uh far but not that far from
57:21well it's usually yes
57:24um just for your information uh you're
57:26you're out of the uh of the screen I
57:28think at least for me
57:31um myself or the what I write so
57:34yourself and actually I can I can read
57:37the 10 but not the exponent of the 10.
57:39okay okay thank you I'm not sure
57:42so are you because if hydro line here
57:45it's okay
57:46it's still a bit too far right
57:50okay so I'll try not to go beyond that
57:53sorry for these technical glitches that
57:56are bound to happen in the first lecture
57:58Okay so
58:00so what I'm saying is that such an event
58:02has a probability 10 to the minus 22
58:05and to give you an idea the number of
58:08seconds since the universe is born is
58:10around 10 to the 16. so if you have an
58:14event that happens according to a
58:17gaussian distribution every second
58:19it would still have less than one chance
58:22in a Million
58:23to uh happen in within a gaussian
58:26distribution so this tells you that in
58:29practice
58:30uh if you have a gaussian distribution
58:32if you describe things with the gaussian
58:33distribution events like 10 Sigma events
58:36uh should actually never happen
58:40um okay so now I see I see the screen
58:43thank you for uh printing this out to me
58:48um okay so you know you if you're for
58:52example if you do risk control risk
58:54modeling in financial markets and use
58:57the gaussian distribution
58:58uh then you're bound to make something
59:02wrong because the 10 Sigma events
59:04actually happen very frequently in
59:06financial markets and so
59:09just knowing that means that a gaussian
59:12distribution won't work for describing
59:14how financial markets evolve but still
59:18you may be surprised to know that this
59:21gaussian distribution for financial
59:23Market is still very much the standard
59:26Paradigm at least when you hear about
59:29financial markets in mathematical
59:31Finance lectures the whole description
59:34is built around gaussian distributions
59:39okay another example is the exponential
59:42distribution or the LaPlace distribution
59:43rho of X row L of X which is Theta of X
59:49Theta is a heavy side function so it's a
59:51zero if x is negative
59:54times exponential of minus Lambda X
59:59so this happens very frequently that the
1:00:01a random variable is distributed as an
1:00:05exponential and here again all the
1:00:07moments are are finite
1:00:12so the mean of this distribution for
1:00:15example is given by 1 over Lambda so
1:00:17there's a Lambda missing
1:00:19or the distribution to be normalized
1:00:22so for example M1
1:00:25is equal to 1 over Lambda
1:00:31okay and so
1:00:33clearly there are there's an infinite
1:00:35family of uh of entail distribution
1:00:39um and I just give you two examples that
1:00:42come up uh frequently and I want to give
1:00:46you examples of fat tail distribution so
1:00:48fat tail distribution will be
1:00:49distribution such that some of the
1:00:52moments diverge
1:00:54so for example
1:00:57I'm going to start with a symmetric
1:01:00distribution uh the student T
1:01:02distribution rho of x
1:01:05is a certain normalization
1:01:09that I won't write out explicitly
1:01:14divided by a squared plus x squared to
1:01:17the power 1 minus 1 plus mu over 2.
1:01:21so for example
1:01:24maybe some of you know the case mu equal
1:01:27one
1:01:28and that's a cushy distribution
1:01:37and the generalization for arbitrary
1:01:39values of mu is called the student T
1:01:41distribution
1:01:49what you see is that when X becomes
1:01:51large so again this distribution is
1:01:54symmetric it's uh it only depends on x
1:01:56squared so if x becomes large either on
1:01:59the positive side or on the negative
1:02:01side
1:02:02asymptotically rho s of x
1:02:05decays as 1 over X the one plus mu
1:02:12so can you still see the MU yes just
1:02:16so so maybe I could
1:02:26yeah okay
1:02:31so it has a what's called a parallel
1:02:33tail
1:02:38foreign
1:02:42so you see that if I try to integrate x
1:02:47to the n
1:02:49multiplied by one of x to the one plus
1:02:52mu
1:02:53then the integral will fail to converge
1:02:56if n is too large
1:02:58and you can quickly check that
1:03:02only
1:03:04moments
1:03:06foreign
1:03:14are convergence
1:03:17so if the moment
1:03:19if the order of the moment n is less
1:03:21than this power load exponent
1:03:24table exponent so I'm going to you'll
1:03:27see uh I'm trying to keep a consistent
1:03:30notation for this uh tail exponent mu
1:03:34I'll always call MU
1:03:36uh
1:03:37this object that appears in the in the
1:03:40tail and if n is less than mu then the
1:03:43moment converges but if n is larger than
1:03:46mu the moment diverges so it means for
1:03:49example that if mu
1:03:51is less than one
1:03:53then even the mean doesn't exist
1:03:57the mean
1:03:59is Divergence
1:04:06and if mu is less than two
1:04:08the variance
1:04:10is Divergent
1:04:16so more generally mu describes the speed
1:04:19at which the power law
1:04:21s off at infinity and if mu is small it
1:04:26folds up very slowly if mu is large it
1:04:29falls quicker
1:04:31actually formally you can show that I
1:04:34mean there's a scaling to to be made but
1:04:37in the limit where mu goes to Infinity
1:04:41there's a way to take the limit
1:04:43correctly such that the student T
1:04:45becomes the gaussian
1:04:52so it's an interesting family of
1:04:53distribution that interpolates between
1:04:56the gaussian in some limits and the
1:04:58cushy distribution in another limit but
1:05:00for all finite values of mu
1:05:03um it has parallel tails and some of the
1:05:06moments diverge so this uh
1:05:11this distinction between mu less than
1:05:14one mu less than two and mu greater than
1:05:162 is going to come up uh in a second
1:05:20when we describe
1:05:22in a few minutes when we describe the
1:05:24central limit theorem and its
1:05:26generalization
1:05:27generalizations and you'll see that this
1:05:30is not I mean for many years even in
1:05:33physics people didn't really pay
1:05:34attention to these distributions because
1:05:36it was kind of
1:05:38a pre-assumption that surely a
1:05:41distribution must have a finite moment
1:05:44or and surely must have a fine adherence
1:05:46but you'll see many examples where this
1:05:48is actually not the case
1:05:51another famous Hollow distribution
1:05:56is the Pareto distribution and I'm going
1:05:59to speak about Pi 2 in a second so rho P
1:06:02of x
1:06:03is equal to Mu
1:06:06x0 to the MU over x to the 1 plus mu
1:06:10for X greater or equal and x0
1:06:15and 0 otherwise
1:06:25so this distribution is for
1:06:28variables that all
1:06:30in general positive greater than the x0
1:06:33is assumed to be positive here
1:06:38so
1:06:40um its origin comes from pareto's
1:06:43description and I'm I'll show you in a
1:06:46second of wealth distributions or income
1:06:48distribution and he showed emporically
1:06:51at the beginning of the 20th century
1:06:53that actually wealth or income is not at
1:06:57all distributed according to an
1:07:00exponential or aggression but actually
1:07:01has fat tails has a parallel tails and
1:07:06that's the traditionally now
1:07:09how we call this this type this
1:07:11distribution here is a it's a python
1:07:14distribution
1:07:15and
1:07:17of course the same discussion I just
1:07:19gave here about the values of new apply
1:07:22for the python distribution which has is
1:07:25in a sense of pure parallel it has
1:07:27nothing no other Behavior than a
1:07:30parallel because either it's a power a
1:07:31pure power law here for X greater than
1:07:34x0 or it's zero
1:07:36whereas the student distribution is only
1:07:37a parallel asymptotically
1:07:42okay
1:07:44so let me give you
1:07:46um
1:07:47more information about
1:07:51such random variables and in particular
1:07:55already something that distinguishes
1:07:58them
1:07:59very strongly is
1:08:02the way the maximum of n random
1:08:06variables
1:08:07behaves as a function of n so I'm going
1:08:10to draw n
1:08:16the values of X so X1
1:08:19to
1:08:21xn okay
1:08:24and one natural question in many
1:08:26contexts is to ask
1:08:27so what's the maximum value of these
1:08:30random variables and I'm going to call
1:08:33capital m n
1:08:35like the max is the max
1:08:39from I equal one
1:08:41to n
1:08:44of the excise
1:08:46okay
1:08:49so this you know the menu circumstances
1:08:52where you might ask what is the largest
1:08:55of n random variables so if you're
1:08:57interested in building a dam for example
1:09:00on the river you may ask about the flaws
1:09:04and the level of
1:09:06of the river every every winter and so
1:09:10if you have an observation of I don't
1:09:12know A Century of data you have a
1:09:15hundred value of the maximum height
1:09:17maximum level of the of the river during
1:09:21a given winter
1:09:23and you might might ask well what's the
1:09:26maximum over the century of of this
1:09:29level and how should I expect
1:09:32the dependence of n to carry on in the
1:09:35future
1:09:36so can I extrapolate and think of what
1:09:40is going to be the maximum level over uh
1:09:43a thousand years for example
1:09:45but there are obviously many other
1:09:49contexts in which you're interested in
1:09:51the max again financial markets if you
1:09:54have a portfolio and you look at the
1:09:57change of value of your portfolio from
1:09:58one day to the next then of course
1:10:01you're interested in knowing uh what
1:10:03what the worst that can happen and so
1:10:06you're going to ask about the maximum of
1:10:09of these uh random variables which are
1:10:12daily returns of your investment
1:10:16and so
1:10:17I'm gonna I'm not going to uh do the
1:10:20math but uh maybe that's the first
1:10:22exercise you can think about how would
1:10:25you prove the result that I'm going to
1:10:27give you
1:10:29um well in the gaussian case
1:10:32when n goes to Infinity
1:10:36m n
1:10:38grows like Sigma square root of 2 log m
1:10:46so you see that clearly the more I
1:10:51pick variables the larger n is
1:10:55the larger I expect the maximum of of
1:10:58these random variables to be because the
1:11:01more I try the more accidentally I may
1:11:04find a large value
1:11:06but you see that in the gaussian case
1:11:08this growth is
1:11:11excruciatingly slow I mean log n is
1:11:14already a slow function here it's square
1:11:16root of log n the way it grows as
1:11:18impossible and of course this
1:11:22the square root of log n reflects the
1:11:24fact that the tail of the gaussian is so
1:11:27quickly decaying it's the same
1:11:29information as what the one I was giving
1:11:32you before if the probability to find 10
1:11:35Sigma is a 10 to the minus 22 event you
1:11:38you must have a very large number of
1:11:40observables in order to see it so the
1:11:43maximum value grows only very very
1:11:45slowly
1:11:47if you look at the LaPlace distribution
1:11:50you find that MN
1:11:54froze as a 1 over Lambda log n
1:12:01so still flow
1:12:02I shouldn't change notation so let me
1:12:05write logs like this
1:12:08so still slow but a little faster than
1:12:11the gaussian
1:12:13now if you look at student or Pareto
1:12:17the growth is much faster and you find
1:12:21that
1:12:23foreign
1:12:25the largest event random variables grows
1:12:27as n to the power 1 over mu
1:12:31note for example that if mu is less than
1:12:34one
1:12:35which as you remember is the case where
1:12:38the mean is Divergent
1:12:41MN
1:12:43grows faster
1:12:48than n
1:12:53so if n is greater than one then the
1:12:56maximum grows slower than the number of
1:12:58terms that you've drawn but if mu is
1:13:01less than one it grows even faster
1:13:09Excuse me yes so are we talking about
1:13:14um because you're writing it it's
1:13:16equivalent to so are we talking about on
1:13:19average or is it uh yes thank you for
1:13:22the question so so this is yeah exactly
1:13:24so thank you I was going to make that uh
1:13:27slightly clearer in a second so this is
1:13:31you know in order of magnitude type of
1:13:33arguments so if you ask okay what's the
1:13:36typical order of magnitude of of MN in
1:13:39these cases then these are the results
1:13:41that you should remember but of course
1:13:43one can be much more precise than this
1:13:45and ask what is the probability key of
1:13:49MN
1:13:50knowing n
1:13:52and uh and you can actually give much
1:13:57more precise characterization of these
1:13:59MNS in particular uh in the gaussian
1:14:02case or in the uh in the LaPlace case if
1:14:07you shift MN by its average value or
1:14:13these these quantities here you find
1:14:16that the the little residual once
1:14:18rescaled has a universal distribution
1:14:20but I don't want to go too much into the
1:14:22details so you can think of these two as
1:14:26the average value of the maximum because
1:14:28of course the maximum won't be the same
1:14:30if you draw another sample of X but if
1:14:34you're if you're interested in the
1:14:36average or the typical order of
1:14:37magnitude this is good enough in the
1:14:39case of parallel variables then it's
1:14:42more subtle because the average itself
1:14:45might not exist so if you want to think
1:14:48of that say as the median uh yeah in all
1:14:53cases maybe the median would be the
1:14:57simplest way to describe what I mean by
1:14:59by approximately equals let's say that's
1:15:02median
1:15:04okay thank you
1:15:08okay so let me give you two more
1:15:11um
1:15:13uh
1:15:16information about about this one is
1:15:21um this question of the maximum can be
1:15:23generalized you can ask something about
1:15:27what's
1:15:29the median value say of the nth largest
1:15:33as a function of n
1:15:36okay
1:15:37so let's uh reorder these random
1:15:41variable X1 x n
1:15:44into y1 which is
1:15:48Max
1:15:50of x i
1:15:52larger than Y2 larger than Etc
1:15:57so y1 is the largest Y2 this is the
1:16:00second largest and so on and you can ask
1:16:03a more detailed question not about the
1:16:06the largest one but
1:16:08the the value of the nth largest one and
1:16:12again you could ask about the full
1:16:14distribution of this object but here
1:16:17what you uh just need to know is that
1:16:21the only thing you need to do here
1:16:24is to divide
1:16:26the little n large n by by Little M
1:16:31and this is true
1:16:33when n
1:16:38is much less than capital M
1:16:42so this simple rule of changing capital
1:16:45N into capital N over small N is a rule
1:16:48of thumb that only
1:16:49Works in this regime when small n
1:16:53Becomes of the order of capital n and
1:16:55it's not as simple
1:16:59okay so what you see what you can see
1:17:02from from these expressions
1:17:05is that if you're interested in the Gap
1:17:08Delta between the largest
1:17:12and the second largest
1:17:16then these distributions uh thin or or
1:17:19fat behave very differently in the sense
1:17:22that
1:17:25for the gaussian or the LaPlace
1:17:27distribution
1:17:28let's say for the gaussian distribution
1:17:30this Gap goes down
1:17:34with n
1:17:39whereas for a
1:17:41student entire to the Gap increases
1:17:45within
1:17:46foreign
1:17:52it remains constant
1:18:04so that's another interesting
1:18:06characterization of these fat tail
1:18:08distribution is that the the difference
1:18:10between the largest and the second
1:18:12largest
1:18:14is bigger and bigger as n increases so
1:18:17there's more and more contrast between
1:18:20the champion and the vice Champion if
1:18:23you want for these parallel
1:18:25distributions whereas for the gaussian
1:18:27distribution is the other way around
1:18:28that when the size increases there's
1:18:33less and less difference between
1:18:35the champion and the second champion and
1:18:38the vice champion
1:18:40foreign
1:18:45ly say something that I'll repeat later
1:18:49or maybe okay I'll I'll skip that for
1:18:52the moment and mention that
1:18:55inappropriate so maybe it's a good time
1:18:58to make a pause
1:19:00so what I propose is that
1:19:03um
1:19:03we reconvene in in 15 minutes
1:19:07and continue
1:19:09uh with uh
1:19:12with what I had to say and and show you
1:19:14data okay is there any question in the
1:19:17chat or
1:19:20so is it big enough to screen now yes I
1:19:22think it is
1:19:26okay so see you in 15 minutes
1:19:55excuse me
1:20:00foreign
1:36:52welcome back everybody
1:36:56all right
1:36:57I hope that with the
1:37:00white uh Cloud here you're still going
1:37:03to see my writing let's see
1:37:06otherwise I'll use water
1:37:10okay
1:37:11um
1:37:12can you still hear me are you around
1:37:21yes okay good
1:37:27foreign
1:37:31this this whole stuff here is to allow
1:37:35you to understand what I'm going to show
1:37:36you uh in terms of empirical data
1:37:41and of course this is also going to be
1:37:43useful from a theoretical point of view
1:37:46but but I'm going to show you are plots
1:37:49that elicit these parallel distributions
1:37:54and so before giving you I'm showing you
1:37:57these plots I need to explain to you
1:38:00what is plotted
1:38:01and so in order to see uh the tale of
1:38:05these distributions
1:38:07because we expect
1:38:09paulos maybe in the Tails but not
1:38:12certainly not everywhere
1:38:14people usually plot two different
1:38:18objects that are actually very closely
1:38:20related so let me start with what people
1:38:23call zip slots
1:38:29after Mr zip
1:38:32who did that for um distribution of the
1:38:36frequency of words
1:38:38in a in a written Corpus
1:38:41for example
1:38:43the number of times the V appears in
1:38:46Ulysses I'll show you data so what you
1:38:49do is
1:38:50you actually
1:38:52rank your data you start ranking the
1:38:55data you so you have a fast
1:38:58observation of unranked data which is
1:39:01the X1 xn
1:39:05and then you rank them
1:39:07and what you do is
1:39:10you plot
1:39:13the log
1:39:15of y m
1:39:18as a function of small n over capital n
1:39:23so it's a sorry of the log of small n
1:39:26over capital n
1:39:28so this is a very natural idea right
1:39:30it's the thing that you would do
1:39:31actually a lot of data is presented in a
1:39:34ranked way so for example if you look at
1:39:37uh the most the hundred most uh wealthy
1:39:41man in the on the planet or uh whatever
1:39:45it's usually already ranked so it's a
1:39:49very natural temptation to say okay what
1:39:51happens if I plot
1:39:54why as of the the amplitude of the rank
1:39:57variable as a function of its Rank and
1:40:00what you find is that
1:40:02because there's noise but if you have a
1:40:05parallel tail
1:40:06then in a log log representation
1:40:09it looks like a straight line and the
1:40:12slope of this straight line is
1:40:15minus one over mu
1:40:17why well it's it's because exactly what
1:40:20you're doing
1:40:21this right
1:40:24and I should add uh
1:40:27okay let me change notation here to be
1:40:29consistent and not erase everything
1:40:32output small n as a second
1:40:40so you see that
1:40:44this result here tells you that as a
1:40:47function of small n this is a function
1:40:49of the rank you expect the amplitude to
1:40:52decay
1:40:53as 1 over n to the one over mu
1:40:56and so if you plot the log of the
1:40:58amplitude y as a function of log n then
1:41:03you get the slope
1:41:04minus 1 over mu
1:41:08um so that's that's one way people
1:41:10present the data there's another which
1:41:12as you'll see is completely equivalent
1:41:14there's another way which is based on
1:41:16the empirical
1:41:21cumulative distribution function CDF
1:41:27so what you do is to try to determine
1:41:32empirically what I always write P
1:41:35greater than as a substrate of X which
1:41:38is by definition the integral from X to
1:41:40Infinity of d y rho of Y
1:41:45okay
1:41:46so it counts so P sub of X is the
1:41:50probability to find an observable larger
1:41:52than x
1:41:54and a way to reconstruct this object
1:41:58from empirical data a very standard way
1:42:01is
1:42:03the following
1:42:06so you put again
1:42:09the largest
1:42:10observable
1:42:12y1 here the second largest here and so
1:42:17on Y3
1:42:18and the empirical
1:42:21distribution CDF
1:42:24is constructed
1:42:27by
1:42:29putting zero
1:42:31for X greater than y1
1:42:33then 1 over n between y1 and 1 2 2 over
1:42:38n between
1:42:39Y2 and Y3 and so on so it's a step
1:42:42function that grows
1:42:45in steps of 1 over n one over capital n
1:42:49every time you encounter
1:42:52an empirical variable and it makes a lot
1:42:55of sense right the probability the
1:42:56empirical probability to find something
1:42:58larger than y1 is zero the probability
1:43:01to find something larger than y two is
1:43:04one so there's one event so it's one of
1:43:07Rand and so on okay
1:43:09so that's the way people uh reconstruct
1:43:12empirical
1:43:15cdfs
1:43:16and there's a long story about
1:43:19statistical testing
1:43:21that is determining whether an empirical
1:43:24distribution is compatible with a
1:43:27theoretical distribution or with an
1:43:29another empirical distribution for that
1:43:32matter
1:43:33and so there's a variety of tests the
1:43:35most famous one is is called The
1:43:38Commodore graph spinoff test
1:43:41foreign
1:43:54test which is quite nice
1:43:57but these tests are all based on
1:44:01um I mean many of these tests are based
1:44:03on on this construction of the empirical
1:44:06distribution
1:44:07anyway if you think a little bit for two
1:44:10seconds you'll realize that this object
1:44:15here
1:44:15is just this one flipped around the x
1:44:19equal y axis so if I if I make a flip of
1:44:23this object
1:44:25by the Symmetry around the 45 degree
1:44:28axis I'm reconstructing exactly a zip
1:44:31plot
1:44:32so it's really a matter of taste and
1:44:35presentation and in particular if you do
1:44:39a long log representation then you'll
1:44:42get a straight line as well but instead
1:44:44of having a straight line of slope minus
1:44:471 over mu in this case the slope
1:44:51will be minus mu
1:44:55okay
1:44:58the reason it's minus mu is because for
1:45:01a parallel tail distribution if you
1:45:03inject rho of Y
1:45:05equals 1 over y to the one plus mu then
1:45:08P larger than or a parallel tail is
1:45:12proportional to
1:45:13x to the minus mu
1:45:17okay this is just the anti
1:45:19antiderivative of one over x to the one
1:45:22plus b so that the minus mu here is the
1:45:24slope okay
1:45:27foreign
1:45:30so before showing you data I want to
1:45:34also justify the fact that parallel
1:45:36Tails parallel distributions are also
1:45:39often called scale free distributions
1:45:55okay and this is important because
1:45:58you'll see many examples where
1:46:01one gets a parallel distribution because
1:46:04there's a physical scale in the system
1:46:06that disappears and uh and and the logic
1:46:12here is that if
1:46:15you look at a parallel distribution and
1:46:18you're interested in the following
1:46:19question what is the probability to
1:46:22observe an event larger than 10 times x
1:46:26relative to the priority to observe an
1:46:28event of size X okay
1:46:32so I'm asking in the case of the Python
1:46:36question the parietal wealth
1:46:38distribution question what is the
1:46:40priority to to observe someone earning
1:46:43or uh
1:46:45owning 10 million dollars compared to
1:46:49the property to find someone who only
1:46:51owns more than one million dollars
1:46:54well
1:46:56the nice property of these parallel
1:46:58distribution is that
1:47:01for a student or paito or whatever
1:47:05parallel distribution this is equal to
1:47:0810 to the minus mu
1:47:10independent of x
1:47:17and that's where this scale free ID
1:47:19comes from is independently of the scale
1:47:22at which you're looking at this
1:47:23distribution
1:47:25relative quantity is relative
1:47:26frequencies are completely independent
1:47:29of the scale
1:47:31of course and I'm going to insist on
1:47:33that in a second when you have a
1:47:35parallel distribution a parallel tail
1:47:38it's only valid empirically within some
1:47:41range and so this idea of scale 3 is
1:47:46valid in the same range where the
1:47:48parallel distribution holds
1:47:51and this should be compared for example
1:47:53to the LaPlace case if you compute this
1:47:55for the LaPlace case you find
1:47:57exponential of minus
1:47:599 Lambda times x
1:48:03which strongly decreases with x
1:48:12so what it means is that
1:48:14given this ratio
1:48:17in one case you cannot determine at what
1:48:19scale the measurement was done
1:48:22whereas in this case you can actually
1:48:24extract from this ratio the scale of the
1:48:27measurement because it still appears
1:48:30so in order to have a kind of vivid
1:48:31illustration of this let me give you the
1:48:34example of Jew figures
1:48:39Jose
1:48:43so as you as you know Jew is uh
1:48:46is water
1:48:49absorbing on on glass for example
1:48:52and so you can with a microscope look at
1:48:55the droplets of water
1:48:58that you see
1:49:00and
1:49:01in some cases you see a pretty uh mono
1:49:06size
1:49:08pattern with a droplets that all have a
1:49:12certain characteristic size
1:49:15and so what it means is that you know
1:49:17with your camera if you zoom too much
1:49:20you won't see anything anymore because
1:49:22otherwise either you'll be between
1:49:25droplets or you'll be within one droplet
1:49:27and so you have to choose the correct
1:49:29scale to look at this picture
1:49:32but there are cases where U is scale
1:49:36free
1:49:37in the sense that there's a
1:49:39superposition of very small droplets and
1:49:41very large droplets okay that coexist
1:49:45and so you have droplets of all sizes
1:49:48and actually if you look at the
1:49:50distribution of size of these droplets
1:49:52the distribution of the radius you find
1:49:54that um it's it's a parallel
1:49:58and in this case what it means is that
1:50:00provided you're in the regime where it's
1:50:02a parallel then independently of the
1:50:05zoom you will always see the same
1:50:07picture so the relative frequency of
1:50:09these very small dots that you don't see
1:50:12and the big drops
1:50:14the big droplets is exactly the same
1:50:17independently of the scale you've you
1:50:19choose to look at the picture okay so
1:50:22this is nice because it's a it's a
1:50:24visual way to think about this idea of
1:50:27scale free with the physical scale
1:50:28that's a geometric scale but this is the
1:50:31idea essentially okay
1:50:34so very importantly if you have a
1:50:37parallel phenomenon it means somehow
1:50:40that a scale has disappeared
1:50:42has disappeared and so this guides the
1:50:46the construction of models uh to to find
1:50:50reasons for The Disappearance of a of a
1:50:53length scale or any other types of
1:50:55skills
1:50:58um at this point let me
1:51:00insist on something that I've alluded to
1:51:04several times now which is that
1:51:08in physics we know about this but in any
1:51:11uh
1:51:13natural science observation or
1:51:16observation in physics in economics or
1:51:19in finance
1:51:21one never expects to have a pure
1:51:24parallel distribution
1:51:26for all X's going to Infinity one always
1:51:31has cut off scales and I'm going to show
1:51:34you uh some examples of that
1:51:37um that appear naturally if you think
1:51:40about the phenomenon so I'm going to
1:51:41show you for example a data on
1:51:44earthquakes the magnitude of earthquakes
1:51:46which is a beautiful parallel
1:51:47distribution
1:51:49but of course if you think a little bit
1:51:51about earthquakes you know that an
1:51:53earthquake is related to
1:51:55[Music]
1:51:56um
1:51:58something happening uh on the on the
1:52:01crust of the earth and that the crack
1:52:04explaining the earthquake
1:52:07the size of the crack is related to the
1:52:09amplitude of the earthquake and clearly
1:52:12the size of the crack cannot be larger
1:52:13than the size of the other planet right
1:52:15so clearly this fallow is not going to
1:52:18hold you know mathematically forever
1:52:20it's going to be cut off at one point
1:52:22and so one has to be very careful when
1:52:25one speaks about parallels is to try to
1:52:28identify the regions either a priori or
1:52:32empirically where you see a power but
1:52:35we'll see models where you can actually
1:52:37explicitly see that there's a parallel
1:52:41in some regime and then there's a
1:52:43cut-off which comes from the the physics
1:52:47or the mathematics of the model and
1:52:49we'll see that in the context of
1:52:51branching processes for example
1:52:53and and we know in these cases that
1:52:56there is indeed a cutoff Beyond which
1:52:58it's futile to look for a parallel
1:53:01because it doesn't exist so this is a
1:53:03very important statement because many
1:53:05times people try to use statistical
1:53:07tests like the cosmograph test to test
1:53:11whether the tail of the distribution is
1:53:13a parallel or not and you may get
1:53:15negative uh answer to your tests simply
1:53:19because you're probing the distribution
1:53:21in a regime where it's not any more
1:53:23parallel so this question of statistical
1:53:25test is very important because what is
1:53:28what you see in the literature is often
1:53:30a blind use of tests without asking the
1:53:33right questions of whether you should
1:53:35use the test or not or whether you
1:53:37should use a test with care so one
1:53:39should be very careful with kind of
1:53:41press the button type tests which in
1:53:45many cases is actually not adapted to
1:53:47the question you are asking so uh
1:53:50here I'm advocating a an element of
1:53:53judgment when you do science
1:53:56rather than you know blindly applying
1:53:59tests that are from a mathematical point
1:54:01of view rigorous and you'll see that the
1:54:03Commodore Gross Middle test provided you
1:54:06you give yourself some hypothesis is a
1:54:08very beautiful test very beautiful
1:54:10because it it's in a sense a universal
1:54:13test
1:54:13but um but on the other hand you
1:54:16shouldn't be fooled by theorems once
1:54:19again and you should use your judgment
1:54:20to know whether it makes sense to use
1:54:22the test or not
1:54:24anyway so this is my ranting about
1:54:27trying to be too rigorous when it's not
1:54:30warranted to
1:54:32apply rigor
1:54:35okay so at this point I think I have
1:54:39enough material
1:54:41to show you uh empirical data
1:54:46so
1:54:52with the quality of
1:54:54the video ah
1:55:03okay uh so I'm told that you have
1:55:05problems with the resolution for me on
1:55:08the on my screen it's perfect
1:55:14so are you sure it's your connection or
1:55:17because I I don't see how I can do it
1:55:20better
1:55:26what do you want me to do
1:55:31this
1:55:39really camera now that's image
1:55:44Yes actually I think the the image
1:55:47quality is okay for some of us so
1:55:50it's probably a matter of connection
1:55:53yeah I think it's a matter of connection
1:55:55we have fiber optics and it's been
1:55:57blurry 50 of the time
1:56:00I see okay I I don't know what to say
1:56:03this doesn't usually I'd wonder if it's
1:56:06not go to meeting when there are a lot
1:56:08of people that I don't know
1:56:10not as good as maybe Zoom
1:56:13well maybe I'm really sorry about this I
1:56:15can't do much because from my side I
1:56:17mean the camera is is focused correctly
1:56:20in uh
1:56:21I don't know the focus is good I mean
1:56:23when it works I I see it well but
1:56:26most of the time it's just blurry
1:56:30okay I'm really sorry about this
1:56:32I hope um
1:56:35with the sound and with election notes
1:56:37you'll be able to uh
1:56:40make for it
1:56:42okay so anyway I'm going to
1:56:46switch to uh my screen now
1:56:52foreign
1:56:55so tell me if
1:56:57you see it correctly now
1:57:08more okay never mind okay so can you see
1:57:12my screen correctly now
1:57:15someone doesn't see it please Shout
1:57:20okay so as promised
1:57:22um I'm going to start by showing you one
1:57:25of the best known
1:57:26uh result on on parallels which is the
1:57:31so-called
1:57:32um Gutenberg rishta uh distribution of
1:57:35earthquakes
1:57:36so this the the right plot here the left
1:57:40plot here that I'm showing with my mouse
1:57:43is
1:57:45um as a function of time
1:57:47the amplitude of the released energy of
1:57:51uh earthquakes that are detected
1:57:54and so what you see is that
1:57:58um there's there are many actually many
1:58:00earthquakes some of them you don't even
1:58:02see
1:58:04and and then some big ones
1:58:06and you can trust me and we'll see uh
1:58:09explicit pictures of that later is that
1:58:12if you zoom for example on this little
1:58:15subpart here you'll see a pattern that's
1:58:17very similar to the big one and actually
1:58:19if you even if you zoom on smaller
1:58:22portions of the time series and you blow
1:58:25it up so zooming meaning uh putting the
1:58:28largest event here on the scale of the
1:58:31full picture you'll again see this uh
1:58:34intertwining of large events and small
1:58:37events okay so this is exactly the scale
1:58:41free uh phenomenon that I was uh telling
1:58:45you about and in fact you can really see
1:58:48that there's a lot of earthquakes that
1:58:50you don't see because if you convert the
1:58:53uh energy radiated by the earthquake
1:58:56into the magnitude of the earthquake the
1:58:59up the cons that people talk about in
1:59:02the news when earthquake happens this is
1:59:05actually showing you earthquakes between
1:59:06seven and eight on the on the Richter
1:59:09Scale so a lot of earthquakes below
1:59:12seven you don't even see or below six
1:59:14even so now if you uh make a histogram
1:59:19of the number of earthquakes as a
1:59:22function of the magnitude
1:59:23which the magnitude being just the log
1:59:26of the of the energy then you see uh the
1:59:30log of the of the number of observations
1:59:33is function at the log of the magnitude
1:59:34so this should have a slope one plus mu
1:59:37minus one plus mu if it's a parallel
1:59:40this is a PDF not a CDF
1:59:43and you see that well between 10 to the
1:59:47minus 4 and 10 so over five decades at
1:59:51least it's pretty nicely uh parallel
1:59:54with the distribution of released energy
1:59:56that decays as e to the minus five third
2:00:00so in my language mu is equal to 2 3
2:00:04here okay and so mu equal two third
2:00:07remember it's 1 plus mu the power of the
2:00:11of the probability distribution of the
2:00:13density and so if mu is two third it's
2:00:16less than one so formally uh the average
2:00:19value of the reduced energy is infinite
2:00:22but of course as I said it in the end
2:00:26this parallel has to stop somewhere
2:00:28because uh otherwise it would mean that
2:00:31you have earthquakes uh the
2:00:34characteristic size of which is larger
2:00:36than the size of the F so we know that
2:00:38somewhere maybe somewhere that you we
2:00:40don't see the distribution should fall
2:00:43off uh more rapidly than a parallel so
2:00:46mathematically the the actual uh mean
2:00:50value of this distribution will exist
2:00:52because of the cutoff but in a in a very
2:00:55wide region of observations you don't
2:00:58see the cutoff and you see the
2:01:01um
2:01:02the parallel
2:01:05what I'm showing
2:01:07um here is uh in a sense power laws in
2:01:10the lab so this is called
2:01:13um buckhausen noise so the way you get
2:01:16back as a noise is you take a magnet
2:01:18with impurities and you try to magnetize
2:01:22the magnet by putting a an external
2:01:24magnetic field and you raise the
2:01:27magnetic field slowly and what you see
2:01:30is that
2:01:31um sometimes so in a magnet there are
2:01:33domains and domain walls and the domain
2:01:37walls are pinned by impurities and
2:01:39prevents the size of the of these
2:01:41domains to grow continuously in
2:01:44principle because you're driving the
2:01:45system with a small continuous increase
2:01:49of the magnetic field you should have a
2:01:52continuous response a continuous
2:01:54increase of magnetization but that's not
2:01:56what you see you see that the
2:01:57magnetization actually
2:01:59is constant for a while and then as a
2:02:02domain wall and pins it
2:02:05this continuously increases and every
2:02:08time it increases the magnet emits the
2:02:11sound and so you can actually hear
2:02:13magnets magnetizing and this is called
2:02:16The Buck has noise so what you see here
2:02:18is a recording of every every time
2:02:21there's a spike
2:02:22um you have
2:02:24an increase of magnetization
2:02:27and so what you see here what is really
2:02:29interesting is that although the
2:02:31perturbation is slow and continuous you
2:02:34increase slowly the the external field
2:02:37the response of the system is
2:02:39intermittent
2:02:41so either it's zero either it doesn't
2:02:43move at all or it moves a lot and so
2:02:46this kind of zero one nature of the
2:02:49signal this intermittent nature of the
2:02:51signal is something that we see in many
2:02:53complex systems so that's another
2:02:56illustration of a famous physical
2:03:00realization of a complex system which is
2:03:02turbulent flows and what you see here is
2:03:07the local
2:03:09um if you want the local
2:03:11scale of the Velocity field
2:03:14and so in a low turbulence flow
2:03:18as a function of time or as a function
2:03:20of space it's roughly uniform the the
2:03:23flow is uh everywhere you look the
2:03:26velocity of the flow is more or less the
2:03:28same but in a turbulent flow you have
2:03:31this this intermittent Behavior again so
2:03:35sometimes the velocity is very low
2:03:38sometimes it's high and there are Peaks
2:03:40turbulent Peaks like little tornadoes
2:03:44that go through the uh the detector here
2:03:46that is extremely intermittent
2:03:50and you see the same uh in financial
2:03:53markets so this intermittency of the
2:03:56activity of a tower and flow is
2:03:58reflected as an intermittency in the
2:04:01What's called the volatility that is the
2:04:02propensity to fluctuate of uh financial
2:04:06markets so what I'm showing here is uh
2:04:09between 1900 and 2000
2:04:12uh the absolute value of the daily price
2:04:16change of uh so-called Dow Jones index
2:04:20which is the index of uh which is an
2:04:24average in skipping the details but
2:04:27roughly it's an average giving you the
2:04:29valuation of the American Stock Market
2:04:32and so you see uh again a phenomenology
2:04:35that's very close to what I've shown
2:04:37before you have huge spikes which
2:04:40corresponds to crisis if you want to
2:04:42crashes and then periods during which
2:04:46the velocities is very low so you see
2:04:49this Big Blob here
2:04:50this Big Blob is what happened after the
2:04:531929 crisis
2:04:55uh the 1929 crisis were a huge drop in
2:04:58the valuation of the American Stock
2:05:01Market but then during 10 years or so
2:05:04the activity was very large with many
2:05:07moves this scale here is in percent so
2:05:10you see the 10 10 means that the stock
2:05:12market has gone up or down by 10 and so
2:05:15you see that during 10 years there were
2:05:17wild uh fluctuations in the stock market
2:05:19and then during the 60s nothing much
2:05:22happened and then again some uh what
2:05:26activity spikes and what's interesting
2:05:28is again you have a kind of skill free
2:05:30phenomenon in the sense that if you zoom
2:05:32on a decade 90 to 2000 or uh on a year
2:05:37or on a month or even on a day you see
2:05:41that the volatility of the market is
2:05:43fluctuating In This Very intermittent
2:05:45way with the region of excitations
2:05:48intertwined with regions of com so it's
2:05:52exactly the same as in a turbulent flow
2:05:54and you can actually push this analogy
2:05:57further
2:05:58so in the left curve here you see the
2:06:03distribution of the change of velocity
2:06:09between two points
2:06:11okay so if you measure the velocity on
2:06:13one point in the flow and the velocity
2:06:15uh then the velocity on in on another
2:06:19point of the flow
2:06:21these are usually different
2:06:22and if you make the difference then make
2:06:24the histogram of this difference you
2:06:26find the this family of curve and what
2:06:29distinguishes these different curves is
2:06:31the scale at which you measure the
2:06:33difference that is if you measure two
2:06:35close by points or if you measure two
2:06:38points that are far away and what you
2:06:40see is that if you measure between two
2:06:43points that are close by
2:06:45you see a histogram that has a fat tails
2:06:49so let me explain why I see we speak of
2:06:52fat tails because here on the x-axis is
2:06:56the velocity the velocity difference and
2:06:58on the y-axis is the log of the
2:07:00probability
2:07:01so remember the gaussian the gaussian is
2:07:03exponential of minus x squared so if you
2:07:05take the log you have an inverted
2:07:07Parabola so that's what you see here for
2:07:11difference of velocities on large scales
2:07:14is pretty much a gaussian but then as
2:07:17you zoom in in a sense you see a
2:07:19distribution that becomes fatter and
2:07:21fatter okay and what's interesting in
2:07:25terms of phenomenology is that if you do
2:07:27the the same experiment
2:07:28for the price difference of the S P 500
2:07:33here it's not the Dow Jones but it's
2:07:35another index then you find that if you
2:07:38measure the difference of prices on
2:07:41relatively long time scales it's it's
2:07:43close to a gaussian but as you move to
2:07:45higher and higher frequency and here for
2:07:47example on The Daily time scale you find
2:07:50a much fatter distributions that
2:07:53resemble what you are seeing in the
2:07:56turbulent flow so I'll go back to that
2:07:58in more details
2:08:00so let me speak about now uh other
2:08:03famous parallel distributions I spoke
2:08:05about the earthquakes but
2:08:10the most ancient one maybe is um is the
2:08:13one of taito that I told you about
2:08:16so this is a zip plot
2:08:18of
2:08:19um wealth with the wealth distribution
2:08:23I think it was in 2005. and uh in here
2:08:28it has changed since and this was Bill
2:08:30Gates and this is Warren Buffett and so
2:08:33on and so you see that the zip plot give
2:08:37you something that's relatively straight
2:08:39on actually several decades with an
2:08:42exponent view which is around 1.5
2:08:46okay so 1.5 is greater than 1 so it
2:08:49means that the average wealth formally
2:08:51exists but its variance is infinite
2:08:55and what's uh striking about this
2:08:59observation is that it was already made
2:09:02um in the in the late
2:09:04um 19th century by uh Wilfredo Pareto an
2:09:09Italian economist who actually here is
2:09:12some of this data we collected on income
2:09:16in Great Britain in Ireland again this
2:09:20is a cumulative distribution now not as
2:09:22if thought but as I've told you it's the
2:09:24same content and you see very nice
2:09:27straight lines
2:09:29um and what title realize is that uh the
2:09:33the these parallels are Universal
2:09:36independent of the country he was
2:09:38looking at and with exponents that are
2:09:41relatively close to one another and so
2:09:45he was mentioning this in his uh in his
2:09:47book
2:09:48speaking about these empirical results
2:09:51these results are most remarkable it's
2:09:53absolutely impossible to admit that
2:09:54there are only a result of chance there
2:09:57must be without doubt the cause which
2:09:59produces the tendency for incomes to lie
2:10:02according a sudden curve the shape of
2:10:04this curve seems to depend to a very
2:10:06small extent on the different economic
2:10:08situations of the countries under
2:10:09consideration because the results are
2:10:12more or less the same in those countries
2:10:14whose economy conditions are varied as
2:10:16those of England Germany Italian towns
2:10:19and even Peru so you see it's quite
2:10:23remarkable because already at this time
2:10:271896 Tito had observed something very
2:10:31counter-intuitive in a sense that is
2:10:33this uh extremely broad distribution of
2:10:36wealth and income
2:10:37that doesn't come naturally from simple
2:10:40economic models and he was also struck
2:10:43by the universality of the results and
2:10:45he was looking for something that
2:10:47physicists are fond of which is to find
2:10:50a common cause to
2:10:52apparently different types of phenomena
2:10:56or observations
2:11:00so just one point about the universality
2:11:03of this exponent 1.5 and we'll go back
2:11:06to that when I tell you about a simple
2:11:09model that generates these parallels is
2:11:11that it's not exactly true that they are
2:11:14constants across countries or across
2:11:16time and here what you see is something
2:11:20that you've probably heard a lot in
2:11:23otherwise you live on in a different
2:11:25planet is the fact that the wealth
2:11:28inequalities in the US has evolved quite
2:11:31uh
2:11:33significantly significantly over the
2:11:36last century so that's the whole work of
2:11:40pkt and others and so here
2:11:44um well don't look at the upper graph if
2:11:46you want but from the data you can infer
2:11:49a value of this exponent new and you see
2:11:53that it's a
2:11:55It's relatively low around 1.5
2:11:59around the 1920s that it increases and
2:12:03remember increasing mu means that the
2:12:05distribution falls off faster so there's
2:12:08less inequalities when mu increases it
2:12:11reaches around two around 1980 and the
2:12:16start of the Dragon years and then it
2:12:18goes back down again to Levels Close to
2:12:21the 20s so this is really what people
2:12:24have in mind when they speak about the
2:12:27increase of inequalities in the US and
2:12:30in in the world more generally
2:12:33yes okay other uh famous parallel
2:12:38distributions which are quite remarkable
2:12:41one is
2:12:42the distribution of City sizes
2:12:45so
2:12:47um here you look at the different cities
2:12:51in the country or or in the world and
2:12:55you rank them according to their size
2:12:57and then you do either a rank plots a
2:13:00zip plots or a
2:13:03CDF as I was explaining but it's the
2:13:05same result it's even more the same
2:13:06result because in this case you find mu
2:13:08equal one so if you remember one of the
2:13:11slope the slope of the CDF
2:13:12representation is one is minus mu and
2:13:15the slope of the
2:13:17zip representation is one minus one one
2:13:20over mu but if mu is one of course the
2:13:22two are the same so the value of mu
2:13:25equal one is special and it's called
2:13:27actually a zip flow or a reason I'm
2:13:30going to say in a second and what you
2:13:32see is that again a very nice parallel
2:13:35distribution for City sizes
2:13:38and if you do the same
2:13:41exercise for Farm sizes in the US you
2:13:45find and look here it's really
2:13:46impressive because you go from sums of
2:13:49size 10 so 10 people working in the farm
2:13:52around here to Farms like Walmart where
2:13:56there's a million employees so there are
2:13:59five decades here
2:14:00over which the distribution appears to
2:14:04be close to perfect
2:14:07Ziploc that is a parallel with mule one
2:14:10okay so it means that you know in in
2:14:14layman terms it means that earthquakes
2:14:17are extremely heterogeneous
2:14:19City sizes are extremely heterogeneous
2:14:21firm sizes are extremely heterogeneous
2:14:23and wealth distributions are extremely
2:14:25attributions and why is that important
2:14:27is because if you try to represent the
2:14:30whole population of firms or of
2:14:33individuals by an average representative
2:14:37guy or an average representative firm
2:14:40then you know it's it's not clear at all
2:14:43that you're not throwing the baby with
2:14:46bath water by neglecting this huge uh
2:14:49variety of farms and the huge variety of
2:14:52wealth and so including these
2:14:56fluctuations these extreme fluctuations
2:14:57in economic models is something that
2:15:00people are trying to do right now and
2:15:03clearly from the data it's it's really
2:15:06important again vehicle one corresponds
2:15:08to the point where
2:15:10the uh the average of the distribution
2:15:12barely exists it's just the point where
2:15:15it's mathematically starts diverging and
2:15:18so if you have a an infinite
2:15:21average firm size how can you represent
2:15:24the whole economy as a single
2:15:27representative um this doesn't look
2:15:29right
2:15:30and of course it is problematic
2:15:34so zif as I told you zip played the
2:15:37exercise of um
2:15:39of of of making histograms of um
2:15:44uh the frequency of a word as a function
2:15:47of its rank
2:15:48so the number of times the word appears
2:15:50in the text is a function of uh of the
2:15:53small n that I
2:15:55um
2:15:56introduced uh in the on the on the board
2:15:59the the rank of the of the word and so
2:16:02here is what V which not surprisingly is
2:16:06the most common one and then you have
2:16:07this beautiful parallel uh for the
2:16:10distribution of of word frequency and
2:16:13again you see that this is relatively
2:16:15independent of the language in which the
2:16:19text is written so Spanish and French
2:16:23and they all show this very uh broad
2:16:27distribution scale free distribution of
2:16:29uh of word frequencies
2:16:32and it's close to Miracle 1 so that's
2:16:36what zip had noticed and so the miracle
2:16:40one case is now called the zip
2:16:42distribution
2:16:45so again interesting to think of models
2:16:48that could explain why these parallels
2:16:51appear and we'll speak about that later
2:16:53there's another empirical data that I
2:16:56won't show because it's uh it's
2:16:58interesting and at the same time a
2:17:01little depressing this is the analog of
2:17:04the zip plot for
2:17:08um the number of citations that are
2:17:10sudden
2:17:12uh paper has
2:17:14so actually this is a cumulative
2:17:16distribution I think
2:17:20no no it's this is a ZIP file anyway so
2:17:24what you see is that again there's a
2:17:26parallel tail with now an exponent view
2:17:29which is equal to two and uh what
2:17:33happens now is is that there are papers
2:17:35that are extremely well cited because
2:17:37they launched a new field or they made a
2:17:40tremendous amount of progress
2:17:43uh but but this Palo also means that the
2:17:46most probable is that papers have
2:17:49received very few citations and if you
2:17:51look indeed that the most probable value
2:17:53of the number of citations is is around
2:17:56one or two which means that most papers
2:17:59are never cited except by the author uh
2:18:02himself or herself so uh this is a kind
2:18:05of again the um strange phenomenon where
2:18:09either the paper is hardly noticed and
2:18:12you've worked for yourself essentially
2:18:14or the paper has a great success and
2:18:18receives many citations
2:18:20again we expect that uh very far out in
2:18:23detail you there should be something
2:18:25else happening because if you think for
2:18:27example of uh Einstein 1905 papers they
2:18:31are not cited anymore they're excited as
2:18:33books or or textbooks and so here we see
2:18:38another reason why a number of citations
2:18:41might be a parallel in some region but
2:18:43then we expect that Beyond some number
2:18:47uh citations become of a different
2:18:49nature
2:18:51so here is a big list of
2:18:54of examples where you you see parallels
2:18:57and the corresponding
2:18:58um
2:18:59uh value of mu so earthquakes I've
2:19:02mentioned already I said Five Thirds so
2:19:05mu is around 2 3.7 uh industrial
2:19:08disasters the amount insurances have to
2:19:11pay after uh
2:19:13whatever fire in in
2:19:19in a
2:19:21ffecting the industrial sites or things
2:19:25like that and here you find a very broad
2:19:27again parallel muco one uh books books
2:19:31are a little bit like citations some
2:19:33some books uh
2:19:36are sold to incredible numbers and
2:19:39others are only bought by by a few
2:19:41friends of the author you equal 0.5 2.5
2:19:44the box office or the the amount of uh
2:19:48of people buying tickets to go see a
2:19:51movie and this is 1.6 if you look at the
2:19:55worldwide uh White
2:19:58web
2:20:00it's um it's the network
2:20:04um and I'll show data of that later on I
2:20:06mean I'll show pictures of that it's
2:20:09also some nodes are extremely connected
2:20:11and have a lot of neighbors and other
2:20:14nodes are very weakly connected
2:20:16so a very broad variety of examples
2:20:18where these parallels appear and we need
2:20:21to understand where they come from what
2:20:23are they what are they telling us what
2:20:25is the the idea of scale-free phenomena
2:20:28indicating on the nature of the
2:20:31underlying mechanism
2:20:33here I'm showing a little bit the analog
2:20:35of
2:20:37of
2:20:38dark housing noise when you take a piece
2:20:41of material and you try to break it
2:20:43before it actually breaks it's going to
2:20:46make some noise and so if you again look
2:20:49at the energy the acoustic energy that's
2:20:54released by the micro crack growing you
2:20:57see a very beautiful parallel so here in
2:21:00a sense this is really an earthquake in
2:21:02the lab with the same phenomenology
2:21:06foreign
2:21:09this is stock markets now
2:21:12so this is the cumulative distribution
2:21:15of uh price changes from one day to the
2:21:19next
2:21:20and you see a quite beautiful straight
2:21:23line which in a log log representation
2:21:26again
2:21:27indicates uh parallel with exponent mu
2:21:30around three
2:21:32and what I'm showing here is that this
2:21:36exponent 3 seems to be extremely
2:21:38Universal it doesn't seem to depend much
2:21:40on what kind of financial object you're
2:21:43looking at so so this is a student
2:21:45distribution with mu equal three fitting
2:21:48the returns of the daily returns of the
2:21:53s p index
2:21:55this is another object that I don't even
2:21:58want to Define here showing the same
2:22:00parallel Tails this is a superposition
2:22:02of the PDF a very different Financial
2:22:05objects that all show this inverse cubic
2:22:08law as it's called that is the one of
2:22:11Rex Cube
2:22:12distribution
2:22:14Decay for the cumulative distribution
2:22:18and this is
2:22:20um
2:22:21uh
2:22:22plot showing the value of this exponent
2:22:25mu across many different Financial
2:22:27contracts so here you would find uh corn
2:22:31for example so it has an exponent
2:22:34slightly larger than three
2:22:37um
2:22:39wheat the SPX that um
2:22:43US Stock Market
2:22:45all sorts of other things gold and so
2:22:49you see that
2:22:50um well maybe there are a few outliers
2:22:53like Swiss franc here
2:22:55or all the Euros which seem to have a
2:22:58slightly higher values of you but
2:23:01overall we would be tempted to say the
2:23:05same thing as Ito said why is there such
2:23:08a degree of universality between all
2:23:10these observations all these
2:23:12observations seem to be compatible with
2:23:14a parallel and the value of mu is uh
2:23:18pretty uh much the same for all kinds of
2:23:22financial instruments except maybe
2:23:24foreign exchange where you see that
2:23:26maybe in the case when you exchange
2:23:29something else happens
2:23:31and one of the hints that something
2:23:33interesting takes place is that now if
2:23:36you look at
2:23:37so the parallel tail means that there
2:23:39are extreme events and if you look at
2:23:42what happened that can explain
2:23:45uh the the strength of this event why
2:23:48why is why was there such a big jump
2:23:50happening that day and what the surprise
2:23:54is that actually many of these jumps
2:23:56seem to come out of nowhere they don't
2:23:59seem to be related to anything that
2:24:01actually happened in the world that they
2:24:02or that particular minute
2:24:04uh and and so it it suggests that the
2:24:09key to understand these this
2:24:11universality
2:24:12is of endogenous nature it's the
2:24:15nonlinear feedback of the market on
2:24:18itself that maybe explains why the
2:24:21emerging phenomenon which is the this
2:24:23probability distribution which has a
2:24:25parallel tail is is due to a kind of uh
2:24:29self-exciting feedback of the market on
2:24:31itself and not the nature of the news
2:24:34that hit the market this is not to say
2:24:36that when there's a big news nothing
2:24:38happens but most of the time the market
2:24:41jumps and nothing has happened and so
2:24:44this is uh this is really a puzzle that
2:24:47one needs to understand and it's related
2:24:49to what I told you at the very beginning
2:24:51one of the most well-known uh anomaly
2:24:54compared to the standard economic theory
2:24:58is the so-called excess velocity of
2:25:00financial prices which as I've already
2:25:02said uh stock prices move by something
2:25:05like two percent up or down every day
2:25:07and this doesn't seem reasonable I mean
2:25:10it doesn't seem reasonable that the
2:25:12actual value of a company changes from
2:25:15one day to the next by such a big amount
2:25:18and you translate it in terms of say if
2:25:22you think of uh I don't know apple or
2:25:24Tesla uh if you convert the two percent
2:25:27in dollars these are enormous amounts
2:25:30and it's very hard to understand why
2:25:32this should be the case
2:25:35okay there are many more unexplained
2:25:38um
2:25:39uh observable
2:25:42um
2:25:43Hollow of observations in economics of
2:25:46Finance I will not uh show all of them
2:25:49but I want to show one that I find quite
2:25:52interesting
2:25:54which is uh the way the fluctuations
2:25:57regress as a function of the size of the
2:26:00of the firm or of an economy
2:26:04so here what I'm showing is
2:26:07the standard deviation of the GDP growth
2:26:11or of the value of the sales growth or a
2:26:15company
2:26:16um so these quantities fluctuate from
2:26:19one year to the next and you can look at
2:26:23at the fluctuations of these quantities
2:26:25so sometimes if a company a firm makes a
2:26:29good year and so its sales increases and
2:26:32then next year it's a bad year it
2:26:34decreases
2:26:35same for countries
2:26:37sometimes the economy is growing so the
2:26:39GDP increases and sometimes you have a
2:26:42recession and the GDP decreases so you
2:26:44can look at at the variation of sales or
2:26:48GDP from one year to the next
2:26:52this will give you random variables
2:26:54and these random variables have some
2:26:56mean which is the mean growth of a thumb
2:27:01or the mean growth of the economy and
2:27:03they also have fluctuations
2:27:04and so what I'm plotting here oops
2:27:11is
2:27:12um
2:27:12the way
2:27:14the uh
2:27:17Sigma that is either square root or
2:27:19would mean square of the fluctuations of
2:27:21the growth either of sales or GDP
2:27:26um depends on the size of the company or
2:27:29the size of the country you're looking
2:27:31at
2:27:32and what You observe and again it's an
2:27:35observation that's remarkable because it
2:27:37covers many decades from
2:27:40uh sales corresponding to a hundred
2:27:43dollars to 10 to the 12 dollars for
2:27:46countries you find that this regression
2:27:49is fairly well described by a parallel
2:27:51with an exponent which is very small
2:27:530.15
2:27:55and what's also remarkable is that
2:27:58you know the firms seem to be a
2:28:01continuation of the countries or vice
2:28:03versa the country seem to be
2:28:05in a sense super firms
2:28:08um that continue the trend that you see
2:28:10at the level of thumbs and actually we
2:28:12know that some Farms are so big that
2:28:16they have a sales that correspond to the
2:28:20GDP of small countries so it's not
2:28:22completely absurd to
2:28:24um to think that there's a continuity
2:28:27between the two problems but actually
2:28:29here you see that indeed the trend is is
2:28:32continued from firms to
2:28:34to countries maybe for very small firms
2:28:36there's something else happening but as
2:28:39soon as the firm becomes substantial
2:28:41then there is this uh slow Decay and so
2:28:45this observation dates back from the mid
2:28:4790s
2:28:48and it doesn't yet have a completely
2:28:53convincing explanation and what you
2:28:56should remember from this graph is that
2:28:59initially people thought that the Decay
2:29:01would be as one over square root of s
2:29:04and the one over square root of s is a
2:29:06is a very simple
2:29:08Central limit type theorem uh that I'm
2:29:11going to speak about now but essentially
2:29:14if you think of a big farm as a
2:29:15superposition of smaller company a
2:29:18smaller departments then each of them
2:29:22fluctuates maybe independently and the
2:29:25aggregation of independent objects often
2:29:27leads to a one over square root of n
2:29:30decrease of fluctuations and we'll see
2:29:33that in the context of the central limit
2:29:35theorem and the same for countries you
2:29:37can think of countries as the
2:29:38superpositions of many different farms
2:29:40and so the GDP of countries should kind
2:29:43of average out the fluctuations of each
2:29:45of them and lead to a standard deviation
2:29:48that decays as one over square root of
2:29:50uh of s but it doesn't it decays much
2:29:53slower so it means that big countries
2:29:55actually have a GDP that fluctuates much
2:29:58too much compared to this naive
2:30:00diversification argument if you want
2:30:03and so this is the analog of what I said
2:30:06in the context of financial markets
2:30:08there's an excess volatility of
2:30:10financial markets but there's also an
2:30:11excess of GDP volatility that's related
2:30:14to the difference between 0.15 here
2:30:17which decays much slower than one over
2:30:20square root of s and leads to very large
2:30:23countries still having a substantial uh
2:30:27business Cycles that's that's the name
2:30:30that economists give to GP fluctuations
2:30:33they call it business cycles and this is
2:30:36called in the literature the small shop
2:30:37large business cycle puzzle because many
2:30:39of these recessions or many of these
2:30:42increase of GDP are not due to a
2:30:46particular shock that one can identify
2:30:49exactly the same as financial markets
2:30:51seem to fluctuate without external news
2:30:55GDP seems to seem to fluctuate uh
2:31:00of course not always and we are we
2:31:02already mentioned this the covid crisis
2:31:04is clearly not an endogenous crisis it's
2:31:07it's imposed by an external shock which
2:31:10is a virus but in many cases the GDP of
2:31:14the country fluctuates and we don't
2:31:16really know why so uh that's that's
2:31:18called the small shock large business
2:31:20cycle puzzle because maybe there are
2:31:21small shocks that we don't see but they
2:31:24lead to an anomalously large fluctuation
2:31:27of the gep
2:31:28Okay so
2:31:30at this stage I'm going to again switch
2:31:33to The Bold and continue with
2:31:38the theory
2:31:40or
2:31:42analytic analytical tools to describe
2:31:46what's going on
2:31:48so we'll probably be here around until
2:31:5012 15 if it's okay for you
2:31:55okay
2:32:02all right
2:32:05foreign
2:32:10so you see quite a number of interesting
2:32:12empirical phenomena that for many of
2:32:15them don't yet have a plausible
2:32:17explanation or convincing explanation or
2:32:20at least an explanation that people
2:32:22agree on
2:32:23so a lot of things to remain to be done
2:32:32so this was you know in my outline I
2:32:35spoke about two types of distribution in
2:32:38one many examples is the thing that I've
2:32:41shown on my screen and now we move to
2:32:44three generalized Central limit theorems
2:32:50and probably will speak about four next
2:32:53week
2:33:10okay
2:33:12three clts
2:33:16okay so remember I have my set of random
2:33:19variables X1 x n
2:33:22and this is drawn According to some
2:33:25density row of x
2:33:29and now something standard that you can
2:33:31be interested in
2:33:33is uh what happens to the sum of these
2:33:37random variables
2:33:40on from I equal one to n
2:33:42of x i
2:33:45and the central limit theorem tries to
2:33:48tell you something about the statistics
2:33:49of this sum provided uh two assumptions
2:33:53are met which were implicit in what I
2:33:56was saying from the beginning I failed
2:33:59to mention it but nobody screams so I
2:34:01guess that everybody implicitly
2:34:03understood what I meant here all these
2:34:05X's are identically distributed
2:34:08according to the same row of X but of
2:34:10course I failed to mention that they're
2:34:12also independent so you're drawing them
2:34:15independently from each other according
2:34:17to the same distribution row of X so
2:34:19this is these are the standard
2:34:21assumptions of the central limit theorem
2:34:23although I'll mention a little later
2:34:25that these these assumptions can be uh
2:34:30extended and and
2:34:33weakened uh tremendously without
2:34:36changing the final result but let me
2:34:39insist on the standard setting of the
2:34:41central limit theorem which is the
2:34:43so-called IID
2:34:46setting
2:34:49so ID means independent and identically
2:34:52distributed
2:34:54According to some row of X okay
2:34:58and so I'm interested here in the sum
2:35:01and you can think of many reasons for
2:35:04being interested in in the sum so for
2:35:07example uh the GDP of a country is the
2:35:11sum of the Productions of many different
2:35:14Farms the total price change of a stock
2:35:19between now and a year from now is going
2:35:22to be the sum of the daily price changes
2:35:25and so on and so forth
2:35:28okay so what can we say about SM
2:35:31well the central limit theorem you all
2:35:35of you know about it and I'm going to
2:35:37not not derive it and prove it but I'm
2:35:42going to tell you what it means and
2:35:44especially what it doesn't mean
2:35:47so I'm going to assume that
2:35:50um
2:35:51mu is greater than two
2:35:54which is a way to ensure that whatever
2:35:57the tail of the distribution
2:35:59uh
2:36:02the first moment
2:36:04is finite
2:36:06and the second moment or the variance
2:36:08which is M2 minus M1 squared is also
2:36:13finite
2:36:15okay
2:36:17so I don't need it to be a parallel I'm
2:36:19just using mu greater than 2 as a kind
2:36:21of shorthand to say these two moments
2:36:25are are finite
2:36:27but again the distribution of X doesn't
2:36:29need to be
2:36:31of a parallel type it can Decay the way
2:36:34it wants I just want these two moments
2:36:36to be finite okay
2:36:38and then the theorem tells you uh the
2:36:42following it tells you that
2:36:45um if I look at SN minus
2:36:49M times n
2:36:51divided by
2:36:54Sigma square root of M
2:36:57okay
2:36:58so if I shift
2:37:00s n by its mean
2:37:03and we scale it by the correct quantity
2:37:05which in this case happens to be 1 over
2:37:08square square root of M so uh we'll see
2:37:11in a second what it means then the
2:37:13probability that
2:37:17this shifted and rescaled object is
2:37:21between two fixed numbers A and B
2:37:25so the probability for this
2:37:29to be between a and b or any A and B
2:37:32finite this tends for n goes to Infinity
2:37:36to uh
2:37:38integral from A to B
2:37:41of TX over square root of 2 pi
2:37:45exponential of minus x squared over two
2:37:49okay so this is a
2:37:51a rigorous way to State what the theorem
2:37:53means it means that
2:37:55for fixed A and B that you've chosen at
2:37:59the beginning and you won't let them
2:38:01evolve with n that's the important point
2:38:03that A and B are independent of n then
2:38:05by shifting and rescaling SN
2:38:08you find a universal distribution for
2:38:11this quantity which happens to be a
2:38:13gaussian
2:38:14and when n goes to Infinity that's the
2:38:16only thing that can happen okay
2:38:19so what is remarkable in a sense is is
2:38:22the universality of this result
2:38:28it's the maybe the simplest example of
2:38:31universality it's whatever row of X you
2:38:34started with
2:38:36I don't have to even say what it is I
2:38:39just need these two moments to be finite
2:38:41and boom I have this Central limit
2:38:45theorem that holds and that tells me
2:38:47that in the end I'm I'm
2:38:50I'm ending up with a gaussian
2:38:52distribution for this variable
2:38:54okay so that's the formal uh statement
2:38:57of the theorem but let's see a little
2:39:00more in details what it means and what
2:39:01it doesn't mean before uh you know
2:39:04running to conclusions that may be
2:39:06unwarranted and even dangerous in some
2:39:08cases
2:39:11so what it means is that if I'm plotting
2:39:13the distribution of s n
2:39:15as a function
2:39:17of of
2:39:19of SM for n finite but large
2:39:24okay so imagine that I'm drawing a
2:39:28million of these variables and summing
2:39:30them
2:39:31sorry I expect yes being out of our
2:39:34scope oh sorry sorry
2:39:36yes
2:39:39sorry
2:39:45it's a very difficult exercise to speak
2:39:47without even knowing whether you are you
2:39:50know you follow whether you're
2:39:51interested whether whatever so
2:39:54bear with me I really the first time I'm
2:39:57doing this and I I don't find it's very
2:40:00comfortable anyway so uh okay I should
2:40:03check all the time on my screen
2:40:07yeah maybe I can actually actually for
2:40:10some reason I'm biased and I never write
2:40:13the left and I write more to the right I
2:40:15don't know if it means anything but uh
2:40:17so let me
2:40:19buys my screen the other way around
2:40:24okay
2:40:25foreign
2:40:34Okay so
2:40:35so what this theorem tells me is that
2:40:37well
2:40:39suddenly something uh
2:40:41happens around the mean MN which I am
2:40:45taking as the origin here I'm centering
2:40:47around MN
2:40:49and then there's a region
2:40:53where the distribution is gaussian and
2:40:56this region
2:40:59is at least
2:41:01Sigma square root of n
2:41:04and so that's what it looks like in the
2:41:06central region
2:41:09but then
2:41:10what happens is that there is a
2:41:13crossover
2:41:15which I'm going to call Delta
2:41:19star
2:41:23which depends on n and actually might
2:41:27not be even
2:41:28exactly the same to the left and to the
2:41:30right that belong Beyond which the
2:41:34distribution is not gaussian anymore
2:41:39and I'll give you explicit examples of
2:41:42cases where one can compute what happens
2:41:44in these Tails regions
2:41:45so
2:41:47so that's the tail region
2:41:52and in in details it's not gaussian
2:41:55actually
2:41:57and it can be anything
2:41:58and so the the kind of uh paradox
2:42:02that
2:42:04is implicit in the central limit theorem
2:42:06and for those of you like statistical
2:42:08mechanics it's very much related to what
2:42:10I was saying earlier about the Paradox
2:42:13of irreversibility is that for any
2:42:16finite n mathematically there's as much
2:42:20information in P of SM than there is in
2:42:24row of X which means that for any finite
2:42:26n
2:42:27you can in principle reconstruct exactly
2:42:30rho of X starting from P of s okay
2:42:35so it seems contradictory right because
2:42:37I'm saying at the same time that for n
2:42:39going to Infinity the P of s is gaussian
2:42:42and Universal
2:42:43but at the same time there's exactly the
2:42:46same amount of information in row of X
2:42:48and in P of s
2:42:49and the Paradox is that this information
2:42:51specific tour of X it disappears in the
2:42:55tails
2:42:56it's more and more lost
2:42:58in these regions that have a probability
2:43:02to be observed that is less and less and
2:43:05so that's where you you hide the the
2:43:09specificity of the problem that you're
2:43:10looking at it's in the tails
2:43:13so what is Delta of n well it depends on
2:43:16the problem
2:43:18Delta star of n
2:43:21this is not Universal
2:43:30and so I'm going to give you a few
2:43:32examples of of what Delta of n looks
2:43:37like
2:43:37uh but what I'm saying here which is
2:43:40really important is that the tail region
2:43:43is not Universal and the position of
2:43:46this Crossover at Delta star from uh at
2:43:49which you cross over from the gaussian
2:43:52to a tail and of course this is blurry
2:43:54that's why I put a wiggly line here it's
2:43:57not a strict value
2:43:59Beyond which you're not gaussian and
2:44:02before which you are gaussian it's it's
2:44:05something that slowly makes you depart
2:44:08from gaussian and becomes appreciable uh
2:44:12Beyond Delta star okay but it's an
2:44:14important order of magnitude to keep in
2:44:16mind to know whether
2:44:18the phenomenon you you want to describe
2:44:21is in the gaussian region or in the tail
2:44:25region where you cannot use the gaussian
2:44:28central limit theorem and therefore you
2:44:29cannot say anything except if you have
2:44:32information on row of X okay
2:44:34so the reason it's important is that
2:44:37again many people use a central limit
2:44:40theorem
2:44:41forgetting that in central limit there
2:44:43is Central the Sea of of
2:44:46CLT is Central
2:44:50and again it means that
2:44:53it only tells you something in the bulk
2:44:55of the distribution and not in details
2:44:57so imagine that you're a I don't know a
2:44:59portfolio manager and you want to
2:45:02understand the extreme risks of your
2:45:04portfolio
2:45:05you might say well I have a lot of
2:45:08Assets in my portfolio a lot of
2:45:10different financial instruments so maybe
2:45:12I can use the central limit theorem and
2:45:15maybe my risk is very small and my risk
2:45:18in detail is very small because uh
2:45:20because of the gaussian decaying very
2:45:22fast but of course this is crazy this is
2:45:24crazy because of many reasons but one of
2:45:27the basic reasons is that
2:45:30the number of objects that you have in
2:45:32your portfolio is never very very large
2:45:34and even if it was large there would be
2:45:37a threshold Beyond which you wouldn't be
2:45:40able to say anything about the tail
2:45:42event so you know invoking the central
2:45:46limit theorem to control 10 events is is
2:45:48just completely meaningless
2:45:51anyway so let me give you uh two
2:45:54examples
2:45:56one example is the case where rho of x
2:46:01equals rho of minus X
2:46:04so symmetric distribution
2:46:07which uh in this case
2:46:11is such that m equals zero of course but
2:46:13that's not the most important aspect and
2:46:17also
2:46:18uh M4 is finite
2:46:23okay so if you want mu is greater than
2:46:274.
2:46:29no no sorry I'm saying saying something
2:46:32wrong here
2:46:34sorry get that there's something wrong
2:46:37you should look at my notes
2:46:40MN are all finite that's that's very
2:46:44important so my next example is going to
2:46:46be uh when what happens if some of the
2:46:49moments are not finite okay so I'm
2:46:51really looking at uh a distribution with
2:46:55fin tails that is symmetric okay so it
2:46:58has thin tails in the sense that all its
2:47:00moments are finite
2:47:02and then in this case what you actually
2:47:04can show is that Delta star of n grows
2:47:09like n to the three fourth
2:47:14okay and so this is not too bad because
2:47:18it tells you that
2:47:19you have a gaussian of width
2:47:22Sigma square root of n
2:47:25but the width over which this gaussian
2:47:28approximation holds is much larger than
2:47:31square root of n it's n to the three
2:47:33fourth
2:47:34and so um the probability to be in the
2:47:37Tails is is quite small because the
2:47:41probability to be in the Tails would be
2:47:43exponential of minus Delta star squared
2:47:46over
2:47:48Sigma squared n
2:47:52so it's exponential of minus uh
2:47:56n square root of n
2:47:58okay
2:48:00so the probability to be in this region
2:48:03decays quite rapidly with n
2:48:06uh as exponential not of minus n but of
2:48:10minus square root of n which is fast
2:48:11enough so the tail events Decay quite
2:48:15quickly and the the width over which the
2:48:17gaussian approximation holds is much
2:48:20larger than the natural width of the of
2:48:22the gaussian itself so it it's it's not
2:48:25too bad in this case
2:48:27but now look at what happens for power
2:48:29law distribution parallel tails
2:48:40so parallel Tails but I've already
2:48:42assumed that mu is greater than two
2:48:47and in this case because otherwise the
2:48:50central limit theorem as it stands does
2:48:52not hold and I'm going to speak about
2:48:53that in in two minutes
2:48:55but even though if mu is greater than 2
2:48:58then what you can show is that Delta
2:49:01star
2:49:02is now square root of n log n
2:49:09so the central limit theorem has a width
2:49:13a natural width which is square root of
2:49:15M and you're only allowed to use it
2:49:19until
2:49:21points that are not square root of any
2:49:24way but square root of n log n away
2:49:27so it's barely larger than square root
2:49:29of n
2:49:31so very quickly actually
2:49:33the gaussian becomes something else
2:49:35and in this case we know exactly what it
2:49:38becomes
2:49:39it turns out that in the parallel case
2:49:41whatever the value of mu
2:49:46these Tails here
2:49:50are exactly the same Tail as you started
2:49:53with from
2:49:58so you started from a row of X that had
2:50:01a parallel tail with an index mu so he
2:50:05repeated for clarity so I'm assuming row
2:50:08of X is X to the minus 1 minus mu
2:50:11I'm summing these X's together
2:50:16because mu is greater than 2 the central
2:50:18limit theorem holds so I know that I'm
2:50:20going to get the central region which is
2:50:22gaussian but very very quickly
2:50:26you know as soon as I'm a little bit out
2:50:28of the square root of n region only log
2:50:30n away square root of login away
2:50:33I I fall on to something else and what I
2:50:36fall onto is the same distribution I
2:50:39started with
2:50:41and so you here you have a also an
2:50:43illustration of what I was saying is
2:50:45that
2:50:46the specifics of the distribution hides
2:50:49in the tails and so in this case
2:50:53the tail is actually the distribution
2:50:55you started with itself actually it's
2:50:58not exactly the same because there's a
2:51:00factor n
2:51:01here but it doesn't really matter the
2:51:04the functional behavior is the same as
2:51:07the one you started with
2:51:09so in the case of parallel tails the
2:51:11central limit theorem is uh is dangerous
2:51:14to use it's formally true
2:51:17because when mu is greater than two
2:51:19Delta star grows faster than square root
2:51:22of n so this theorem will be valid if
2:51:25you fix a and b at the end the Tails
2:51:29will be expelled far away and so you you
2:51:32will get the central limit theorem but
2:51:34in Practical applications it it's it
2:51:38unless n is enormously large you will
2:51:42always be confronted with this problem
2:51:44of Tails
2:51:46kicking in not very far outside the
2:51:50central region okay
2:51:53so that's what I want to tell you the
2:51:55central limit theorem is well known but
2:51:57maybe these stories about the
2:51:58limitations are not as well known and
2:52:01but still they are very important to
2:52:04keep in mind because in Practical
2:52:06applications they can be uh extremely
2:52:11um
2:52:12detrimental to what your
2:52:15to your objective
2:52:32Okay so
2:52:35so that's good so you already knew all
2:52:38this
2:52:39so something that's maybe less
2:52:41well-known is what happens when you
2:52:44is strictly less than two
2:52:48okay then in this case
2:52:51uh Sigma or even M can be infinite and
2:52:55clearly if
2:52:58if either M or Sigma
2:53:01or actually if m is infinite Sigma is
2:53:03infinite two but if either of those is
2:53:05infinite then this expression is
2:53:08meaningless and clearly something must
2:53:11happen okay
2:53:13so one has to invoke now
2:53:17a generalized Central limit theorem
2:53:19which is due to
2:53:20olivi
2:53:26and actually Central in this case is a
2:53:29little bit of a misnomer and you'll see
2:53:31why
2:53:33so
2:53:35Levy and
2:53:38nidenko
2:53:41in the 30s
2:53:44foreign
2:53:46that will tell you what how you should
2:53:48generalize these uh theorems in that
2:53:52case
2:53:53but before doing this I just want to
2:53:56mention something that maybe some of you
2:53:58are wondering
2:54:00is that I'm insisting on separating
2:54:03mu strictly greater than two
2:54:06here from you strictly less than two
2:54:10here
2:54:11and I'll further divide the interval
2:54:13from 0 to 2 into mu strictly greater
2:54:17than one or strictly less than one
2:54:20so as a as a side remark
2:54:25what about the cases mu equal to
2:54:28mu equal one
2:54:31well
2:54:32they're actually by continuity you will
2:54:34understand what happens in these cases
2:54:36too but there are technical difficulties
2:54:40with these special cases you need to
2:54:43introduce other logs and things like
2:54:45that and I don't want to you know give
2:54:47all the sub cases so
2:54:51it's you know it's it's not a
2:54:54it's not a big deal to just forget about
2:54:56these special cases for now and if you
2:54:59really have to deal with them the
2:55:01results are available as well and they
2:55:03are not very different from the ones
2:55:04that I'm going to speak about
2:55:07okay so what does the central limit
2:55:09theorem of Levy tell you well it tells
2:55:12you that first you should again
2:55:14distinguish between the case where mu is
2:55:18less than two but greater than one in
2:55:21which case m is finite
2:55:25and one is finite so
2:55:28from the case where mu
2:55:30is less than one
2:55:32where even M1 even the mean is infinite
2:55:37okay
2:55:39so if mu is great is between 1 and 2 the
2:55:42central limit tells you the following it
2:55:44tells you that now SN
2:55:47should be written as m n
2:55:50plus u n to the one over mu
2:55:55okay
2:55:56and when n goes to Infinity
2:55:59the distribution of U
2:56:02tends to uh
2:56:05what's called a levy stable distribution
2:56:10Maybe
2:56:12stable
2:56:14distribution
2:56:20which is the analog of the gaussian when
2:56:24mu is greater than two
2:56:25so I introduced two indices here to
2:56:29describe the levy distribution
2:56:31one
2:56:33is Mu is the exponent of the tail itself
2:56:36and the other is beta and beta is called
2:56:40the asymmetry parameter
2:56:43so let me explain what beta is
2:56:46so I'm assuming here that rho of x
2:56:50decays when X goes to plus or minus
2:56:54infinity
2:56:55as C plus minus over X
2:56:59to the OnePlus mean
2:57:01okay
2:57:03so I'm assuming a parallel tail but the
2:57:06amplitude this parameter here is called
2:57:08the amplitude the tail amplitude so the
2:57:10amplitude of the tail doesn't
2:57:12necessarily uh is not necessarily the
2:57:15same to the right or to the left you
2:57:18might have a distribution that's fat
2:57:19tail to the right and thin tail to the
2:57:22left of Vice Versa or whatever
2:57:25so for example even if you have in the
2:57:28case you have a an X that is a positive
2:57:31random variable then the left tail
2:57:33doesn't even exist the distribution is
2:57:35zero so formally C minus would be zero
2:57:38in that case Okay so this is the general
2:57:40assumption that you have Halo Tails both
2:57:43in the left and in the right and beta
2:57:46is simply C plus minus E minus over C
2:57:50plus plus C minus
2:57:53so it's an asymmetric parameter telling
2:57:56you how much uh
2:57:59lopsided is the distribution to the left
2:58:02or to the heavy side and it's to the
2:58:04left compared to the right or vice versa
2:58:06okay so in the case of symmetric
2:58:08distributions C plus equals c minus and
2:58:11beta is zero
2:58:14Okay so
2:58:16you have a family of distributions which
2:58:18are called The Levy stable distributions
2:58:20and they are not known explicitly in
2:58:23general there's no I can't write like
2:58:25the gaussian an explicit formula
2:58:28but there are things that we know about
2:58:30these Levy distributions and one of them
2:58:33is that LMU beta of U times when you you
2:58:40goes to Infinity
2:58:43to
2:58:44um
2:58:46something that
2:58:48well let's say that U goes to plus
2:58:50infinity to simplify it goes to C plus
2:58:54over U to the OnePlus U
2:58:58so
2:59:00you remember I told you that in the
2:59:02central limit theorem case
2:59:05if you stop by a parallel you recover a
2:59:08parallel in the Tails and the tail
2:59:10slowly disappear
2:59:11and the parallel is exactly the same as
2:59:13the one you started with
2:59:17in the case of the levy theorems
2:59:20you keep a parallel forever because this
2:59:22is the limiting distribution
2:59:27is when n goes to Infinity
2:59:30and the parallel is the same as the one
2:59:32you started with okay
2:59:34in the middle there's something
2:59:36different that can happen but the tails
2:59:39are actually preserved so that's why
2:59:43C in this case is not really adapted
2:59:45because in this case it's really the
2:59:48tails that survive and that specify the
2:59:51distribution to which your converging to
2:59:56last thing before going to the case mu
2:59:58lesson one you see that when mu goes to
3:00:022
3:00:07one over mu goes to one half
3:00:11and LU beta
3:00:13converges to a gaussian one can show
3:00:15that that Lu beta for any value of beta
3:00:18and mu beta becomes gaussian
3:00:21so n to the one over mu becomes square
3:00:23root of n
3:00:24and LU beta becomes
3:00:29the gaussian
3:00:31and so you you have a kind of seamless
3:00:34transition between the case of Levi
3:00:38and the gaussian central limit theorem
3:00:40where this in the central limit theorem
3:00:42this statement here with square root of
3:00:44n instead of the of n to the 1 over mu
3:00:48is exactly the same statement as this
3:00:50one here
3:00:51okay you see that if I subtract MN from
3:00:55SN and divide by square root of n I'm
3:00:57left with the random variable U which is
3:00:59the thing that I was interested in both
3:01:05okay I'll come back to the one over mu
3:01:07in a second but uh before doing that let
3:01:10me Express what happens in the case uh
3:01:14mu lesson one well in the case lesson
3:01:17one there's not even this term now and
3:01:19in the case and and mu less than one
3:01:23what you have to write is that SN is u n
3:01:26to the one over mu
3:01:30and again this here the same thing holds
3:01:33that is peer View
3:01:37tense when n goes to Infinity
3:01:39to a lady distribution a levy stable
3:01:43distribution
3:01:45but now its index will be less than one
3:01:47and that's the only thing that changes
3:01:51okay so these are the
3:01:55the statements the statements are if you
3:01:58shift and rescale correctly
3:02:01and in the case mu lesson one you don't
3:02:03even have to shift you just have to
3:02:04rescale then
3:02:07the little variable that remains once
3:02:11shifted and rescaled converges for large
3:02:14n to Universal distribution
3:02:16which in the gaussian case is completely
3:02:18independent of rho and in the lady case
3:02:21depends on the Tails and only on the
3:02:24Tails of row okay
3:02:26you see again these are not as universal
3:02:29as the gaussian which doesn't depend on
3:02:31anything
3:02:32but you you're left with two parameters
3:02:36that describe
3:02:37the Tails of rovex
3:02:41one parameter describes the functional
3:02:43dependence as a function of x the speed
3:02:46of the Decay mu and the other parameter
3:02:48describes the asymmetry of the
3:02:52of the tail amplitudes okay
3:02:57great now I want to
3:03:00tell you a last remark before
3:03:03um stop sorry and yes
3:03:07I have a question about uh this part so
3:03:11this holds only for
3:03:13um distributions that are exactly power
3:03:15laws or for anything that has power loss
3:03:19tails
3:03:20yeah okay so again I'm trying to
3:03:22simplify you right but you could
3:03:25actually have what's called uh slow
3:03:27functions multiplying these these
3:03:30parallels and the theorems would still
3:03:32hold so for example a slow function is a
3:03:35log logarithm log is a slow function so
3:03:38I could put a log here
3:03:41if you want
3:03:43and it wouldn't change the the final
3:03:45result okay but so what is important is
3:03:50the the structure of the power law in
3:03:52the Tails but if the if the power law is
3:03:55uh Dressed with some uh slow function
3:03:59the notion of slow function can be
3:04:01formalized uh completely but think of a
3:04:05log as a slow function or log to some
3:04:06power then uh you're you'll still in the
3:04:10on the safe side okay
3:04:13was that your question
3:04:15uh yes yeah I was a bit wondering about
3:04:18what you said before about the
3:04:20information uh going uh in the tales uh
3:04:25so like where will that information go
3:04:28but that's maybe okay so so that's why I
3:04:30insisted on the fact that this is a
3:04:32really uh kind of flipped situation from
3:04:36the central limit theorem in the central
3:04:38limit theorem that the information is
3:04:40hidden in details but in the in the
3:04:44generalized lady case the information in
3:04:47details is the same as the one you
3:04:49started with so the information is
3:04:51is in the bulk is diffused in the bulk
3:04:54it's hidden in the way you converge to
3:04:57the asymptotic distribution but not not
3:05:00in the Tails everywhere
3:05:02okay thank you
3:05:06um
3:05:07okay so the last remark I wanted to tell
3:05:09you and it's too bad I have to erase the
3:05:11Blackboard was
3:05:13so I'm going to erase just a little bit
3:05:15of Blackboard just to tell you what I
3:05:18want to tell you and expand on that
3:05:20later
3:05:24so you know roughly speaking SM
3:05:29if I for Simplicity I'm going to think
3:05:32of symmetric distribution such that m is
3:05:350
3:05:36but you know I know that upon a shift
3:05:42I can always
3:05:44go back to that case
3:05:47but just for Simplicity
3:05:52what these Central limit theorem tells
3:05:55you is that SN is of all the square root
3:05:58of n when mu is greater than two
3:06:01and S N is of order n to the 1 over mu
3:06:06when mu is less than two okay
3:06:10and because mu is less than two you see
3:06:12that one over mu is greater than square
3:06:14root of n
3:06:15so the spread of SN grows faster with n
3:06:21then square root of n when mu is less
3:06:23than 2. so when mu is less than 2 N to
3:06:26the 1 over mu
3:06:28is much greater than square root of n
3:06:33and this is you know expected in a sense
3:06:36it's because you have these
3:06:38broad distributions so sometimes you'll
3:06:41have a very big event and therefore you
3:06:44expect that the spread of the
3:06:45distribution is going to go faster
3:06:47because you have uh um extreme events in
3:06:51your time series or in your series of X
3:06:54but actually if you remember I told you
3:06:57already about n to the one over mu
3:07:00and that was in the context of the
3:07:02maximum of n random variables and I told
3:07:05you that actually mm
3:07:07is also
3:07:09of older and the one over mu
3:07:13but this time for any value of mu okay
3:07:17and so what you see is that
3:07:20and again this is what I'm going to
3:07:22expand on in detail next week
3:07:25is that in the case mu greater than 2
3:07:31SN
3:07:33is much larger than MN
3:07:36okay
3:07:39for example take mu equals three
3:07:41then the largest of n random variables
3:07:44grows like n to the one-third
3:07:46but the sum goes like square root of n
3:07:49and so asymptotically the sum is much
3:07:53bigger than any of its terms okay even
3:07:56the maximum is small compared to the sum
3:07:59so you're in a kind of democratic regime
3:08:07quote unquote of course
3:08:09where everybody contributes equally more
3:08:13or less to to the sum and there's no big
3:08:16outlier I mean there are outliers but
3:08:18they they remain negligible on the scale
3:08:21of the phenomenon as a whole okay
3:08:24but you see that it's completely
3:08:26different and this is where it's
3:08:28interesting in the case new lesson two
3:08:31because in this case SN is of the same
3:08:34order as MN
3:08:37so the whole sum
3:08:40in a sense
3:08:41as the same
3:08:44amplitude as just one guy
3:08:48and so in this case you have a complete
3:08:51breakdown of this Democratic
3:08:54representation because one guy dominates
3:08:57completely the whole sum
3:09:00and so this you know if you remember
3:09:01this goes hand in hand with the remarks
3:09:04I gave you about distribution of thumb
3:09:06sizes and the idea of a representative
3:09:09firm uh or representative agent if you
3:09:13want to describe the whole population by
3:09:15its average then you better not be in
3:09:18such a situation where a single guy
3:09:21which you know has a lot of wealth or a
3:09:25single firm which is very big actually
3:09:27dominates the whole
3:09:28phenomenon so in physics terms
3:09:32we'll speak about
3:09:34delocalize sums in this case where the
3:09:37sum is delocalized among all its cons
3:09:40components
3:09:42and localized
3:09:44in this case where the sum is actually
3:09:46concentrated in uh in in a few terms and
3:09:51so
3:09:52I'll call this concentrated
3:09:55and will speak a lot about
3:09:56concentrations in the next lecture
3:09:58because this is a very very important
3:10:01phenomenon that appears both in physics
3:10:05but also in economics and so I want to
3:10:08spend some time speaking about this
3:10:10General phenomenon that I will then
3:10:12illustrate with the models that lead to
3:10:16a concentration transition a transition
3:10:18between the two regimes that appear as a
3:10:21function of the values of the parameters
3:10:23if you want Okay so
3:10:25I'm done for today
3:10:28um next week we'll take a a normal
3:10:31Rhythm hopefully with no technical
3:10:34glitches
3:10:35at the beginning so I'll speak as I said
3:10:37uh between nine and ten forty five then
3:10:4115 minutes break and Valentina you have
3:10:45to decide whether you want to spend the
3:10:47narrow more I would favor you know an
3:10:50hour on the 15 at least but you'll be
3:10:53you'll do as you please and in the
3:10:57meantime I'm again very sorry not to be
3:10:59facing you for real because I think it's
3:11:01much more fun that way
3:11:02and I have no idea whether you find a
3:11:05rhythm okay whether it's too slow too
3:11:08fast I have no real questions during the
3:11:11lecture so it's it's uh
3:11:13it's very bothering but I guess that I
3:11:16have to get used to it in and you're
3:11:18already more used to it than I am so um
3:11:21anyway I don't know if I have things in
3:11:23the chat
3:11:38you see jokes
3:11:46okay
3:11:52no remarks
3:11:58yeah I'm clearly here for you to
3:12:00understand and and enjoy so please give
3:12:03feedback feedback for me to go in the
3:12:06right direction
3:12:16okay
3:12:21okay well take care and
3:12:24see you next week
3:12:27thank you goodbye
3:12:30thank you bye
3:12:34thank you see you next week

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