Chapter 7: Mechanisms in Programs and Nature

Section 6: The Phenomenon of Continuity

Related results [to Central Limit Theorem]

Gaussian distributions arise when large numbers of random variables get added together. If instead such variables (say probabilities) get multiplied together what arises is the lognormal distribution

1/(Sqrt[2 π] x σ) Exp[-(Log[x]-μ)2/(2 σ2)]

For a wide range of underlying distributions the extreme values in large collections of random variables follow the Fisher–Tippett distribution

1/β Exp[(x-μ)/β] Exp[-Exp[(x-μ)/β]]

related to the Weibull distribution used in reliability analysis.

For large symmetric matrices with random entries following a distribution with mean 0 and bounded variance the density of normalized eigenvalues tends to Wigner's semicircle law

2/Pi Sqrt[1-x2] UnitStep[1-x2]

while the distribution of spacings between tends to

(π x)/2 Exp[-Pi/4 x2]

The distribution of largest eigenvalues can often be expressed in terms of Painlevé functions.

(See also 1/f noise on page 969.)

From Stephen Wolfram: A New Kind of Science [citation]