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
Exp[-(Log[x] - μ)2/(2 σ2)]/(Sqrt[2 π] x σ)
For a wide range of underlying distributions the extreme values in large collections of random variables follow the Fisher–Tippett distribution
Exp[(x - μ)/β] Exp[-Exp[(x - μ)/β]]/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
2Sqrt[1 - x2] UnitStep[1 - x2]/π
while the distribution of spacings between tends to
The distribution of largest eigenvalues can often be expressed in terms of Painlevé functions.
(See also 1/f noise on page 969.)