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/π Sqrt[1-x2] UnitStep[1-x2]
while the distribution of spacings between tends to
(π x)/2 Exp[-π/4 x2]
The distribution of largest eigenvalues can often be expressed in terms of Painlevé functions.
(See also 1/f noise on page 969.)