Nassim Taleb On Some Consequences Of LLN In The Real World

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Nassim Taleb

Nassim Taleb – Simplest Forms


  • 1-2 Parameter: Minimum value, tail (Pareto I)
  • 3 Parameters: location m, scale s, tail Alpha (Pareto 2-Lomax)
  • 4 Parameters: Generalized Beta 2nd kind (includes many known distributions such as Singh-Maddala, etc.)


  • Student T distribution (finance papers galore)
  • Levy-Stable (to which all those with Alpha<2 converge)
  • Other (double Pareto, etc.)

Sample Equivalence

Dispersion of outcomes: for exponent <2, we cannot use standard deviation and other tools since no second moment. Only MAD , mean absolute deviation of the mean from “true” mean (or 0).

Nassim Taleb


Indexing by p for powerlaw and g Gaussian:

Nassim Taleb

Back door working

  • Sample mean (“realized a average” in language of finance) is never Gaussian when Alpha<2 (even when >2, another story on CLT (following chapter in Silent Risk))
  • Tail Alpha can be estimated with MLE
  • Tail Alpha from MLE is asymptotically Gaussian (preasymptotically Inverse Gamma with low variance reaches v. quickly)
  • We can fit GPD or EVD with same tail exponent Alpha, further reducing variance.

Nassim Taleb

A few points

When 1<Alpha<2 we can safely say that the sample mean is insignificant and underestimates the mean for one-tailed distributions (reason: infinite skewness). For all sample size.

In some cases, with Alpha<1, we can extract the “true” mean, albeit stochastic. Case study on violence (Cirillo and Taleb, 2015).

Application: the Pikery craze.

Centile contribution: is Pareto 80/20 true?

80/20 by recursing -> Top 1% has 53%

Huuuuuuuuge bias in mean measurement as bracketed -> y-o-y changes suspicious

Nassim Taleb


  • Much of finance, social science, relies on bogus estimators. For instance Pinker’s problem is quite insidious with mechanistic users of statistics.
  • We “recalibrate” models because they are not estimators, chasing past fitness. – As the late Benoit Mandelbrot said: when a lightning hits we do not change the laws of nature.
  • Excellent news: rigorous methods, including using extreme value theory and development of new estimators clears up a lot of problems

See full slides below.


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The post above is drafted by the collaboration of the Hedge Fund Alpha Team.

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