The Black Swan: The Impact of the Highly Improbable · chapter 2 · id the-black-swan-c2-08-talebs-mediocristanextremistan
“Taleb's Mediocristan/Extremistan distinction maps onto the mathematical distinction between thin-tailed distributions (Gaussian, exponential) where the Law of Large Numbers converges quickly, and fat-tailed distributions (power law, Cauchy) where convergence is slow or nonexistent.”
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University of Texas: An Example Where the CLT Failssource alive
The Cauchy distribution has infinite variance and undefined mean. Sample averages of Cauchy random variables do not converge -- the average of a million Cauchy samples has the same distribution as a single observation.
Clauset, Shalizi & Newman (2009), SIAM ReviewDOI registry: valid
Power-law distributions have heavy tails, meaning extreme events occur with much higher probability than Gaussian models predict. For distributions with exponent alpha <= 2, variance is infinite.
Mandelbrot (2001), 'Scaling in Financial Prices', Yale Mathematicscheck errored · retry queued
Financial price changes exhibit scaling behavior and fat tails inconsistent with Gaussian assumptions. The distinction between 'mild' and 'wild' randomness maps directly onto the convergence properties of the underlying distributions.
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