The Black Swan: The Impact of the Highly Improbable · chapter 2 · id the-black-swan-c2-04-the-central-limit-theorem-whic
“The Central Limit Theorem, which underpins much of conventional statistics, requires finite variance. When variance is infinite -- as with Cauchy or Levy-stable distributions -- the CLT breaks down entirely, and averages do not converge.”
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Wikipedia: Central Limit Theoremsource alive
The classical CLT requires that the random variables have a finite variance. The Cauchy distribution, which has infinite variance, does not satisfy this requirement, and sample means of Cauchy-distributed variables do not converge to a normal distribution.
University of Texas Mathematics: An Example Where the Central Limit Theorem Failssource alive
The Cauchy distribution does not have a finite variance -- in fact, the Cauchy distribution does not even have a finite mean. The sampling distribution of the mean does not stabilize regardless of sample size.
Mandelbrot (2001), 'Scaling in Financial Prices: Tails and Dependence', Yale Mathematicscheck errored · retry queued
Financial returns exhibit Levy-stable distributions with infinite variance, where the parameter alpha is typically between 1.5 and 2, meaning the CLT convergence is either absent or extremely slow.
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