Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications



  1. Jason ThorellSeptember 3rd, 2020 at 02:00 am


    I have added 15 new edits to the comments in the attached pdf all under today's date, 9/2/20. The prior comments still exist in this file. Hopefully you can sort the comments by date to easily see the new comments. If not, I am happy to write them out similar to how other commenters have done. Feel free to let me know if you need anything.

  2. Ulrich AtzAugust 31st, 2020 at 11:37 pm

    Maybe these have been corrected already:

    • p. 8 "sum of the random variables √ n(Xn µ)" missing the minus sign.
    • erf definition on p.6, 23: √2 ∫ {z 0} dt e^ −t 2  - the integral symbol dt is inside
  3. Nassim Nicholas TalebAugust 30th, 2020 at 04:35 pm

    And the mathematical errata are at

  4. Nassim Nicholas TalebAugust 30th, 2020 at 04:14 pm

    Thanks all! I am updating and adding your names.

  5. Jason ThorellAugust 27th, 2020 at 02:32 pm


    If helpful, please see the pdf file for corrections. I added comments so if you enable the "Comments" feature in Adobe a seperate window will list all of the comments. I hope this is helpful.

  6. Petri HeloAugust 13th, 2020 at 06:01 pm

    Many thanks for providing this great book available on-line. Below you will find few possible typesetting errata:

    p. 37 - These point is explored in the next section here and in an entire chapter (Chapter ??):

    p. 39 - Figure ?? shows the extent of the problem.

    p. 44 - E.E. Slutskii - I am not sure about correct translitterations but on Econometrica article he has used "E Slutzky"

    p. 209 - From figure ?? we can see that, in the equation

    p. 225 - be on average (under some operation of the law of large number deemed satisfactory) [152], [143], see Fig. ??.

  7. Gottfried MundaJune 12th, 2020 at 02:34 pm

    Thanks a lot for making this interesting work freely available!

    It seems that "convexity" is too broad a concept for many of the points made in the book. For example, linear functions are convex, but many arguments rely on the assumption that the function in question is actually growing (nonlinear) in a suitable way. Similarly, many arguments require that equality in the less-equal of Jensens ineq is NOT attained. Thus important classes of convex functions are actually not suited for the arguments in the book (eg linear function, (scaled) absolute value function, indicator function of a convex set: 0 in the set and infinity outside)

    A more precise way would be to replace "convexity" by "strict convexity" (less-equal in Jensens ineq turns into strict less), or, even better, strong m-convexity where m is the parameter of strong convexity. Strong convexity ensures that there's a quadratic lower bound on the growth of the function, which seems to be an assumption for many of the arguments.

    With strongly convex functions it would also make sense to classify one function as being "more convex" than another - the measure is simply the parameter m. In general a function is either convex or it is not, there is no such thing as "more convex" eg p.58

    On p.76 in the quote there's a problem with the typesetting of sigma

    p.77, footnote 6: typo "big ambiguous"

  8. Nassim Nicholas TalebMay 31st, 2020 at 03:56 pm

    Added your name

    to thanks.

  9. Nassim Nicholas TalebMay 31st, 2020 at 03:23 pm

    Thanks. New version is cleaner/

  10. Caio VinchiMay 23rd, 2020 at 05:31 pm

    Mr. Taleb, first thank you for all your work.

    I'm not sure those are proper corrections for the printing version, but I happen to find a few missing links/references on pages:

    pg 136 - Figure 7.10 - "Gaussian Control for the data in Figure ??."

    pg 180 - after equation 10.1 - "As seen in Figure ??, drawdowns..."

    pg 221 - first paragraph - "Misunderstanding g Figure ?? showing..." and "Figure ?? shows a more complicated..." -also missing the page number*****

    pg 283 - last phrase - " values of a < 1, as seen in Figure ??."

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The book investigates the misapplication of conventional statistical techniques to fat tailed distributions and looks for remedies, when possible.
Switching from thin tailed to fat tailed distributions requires more than "changing the color of the dress". Traditional asymptotics deal mainly with either n=1 or n=, and the real world is in between, under of the "laws of the medium numbers" --which vary widely across specific distributions. Both the law of large numbers and the generalized central limit mechanisms operate in highly idiosyncratic ways outside the standard Gaussian or Levy-Stable basins of convergence.
A few examples:
+ The sample mean is rarely in line with the population mean, with effect on "naive empiricism", but can be sometimes be estimated via parametric methods.
+ The "empirical distribution" is rarely empirical.
+ Parameter uncertainty has compounding effects on statistical metrics.
+ Dimension reduction (principal components) fails.
+ Inequality estimators (GINI or quantile contributions) are not additive and produce wrong results.
+ Many "biases" found in psychology become entirely rational under more sophisticated probability distributions
+ Most of the failures of financial economics, econometrics, and behavioral economics can be attributed to using the wrong distributions.
This book, the first volume of the Technical Incerto, weaves a narrative around published journal articles.


➤  Version 1 (2020-01-29)


Nassim Nicholas Taleb (2020). Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications. Researchers.One,

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