Wigner semicircle distribution - Wikipedia

Abstraction: Limiting eigenvalue distribution of large random symmetric matrices

Key points:

• The Wigner semicircle distribution is supported on [-R, R] with PDF shaped as a scaled semicircle; parameter R is the radius
• It arises as the limiting eigenvalue distribution of large random symmetric matrices as dimension approaches infinity
• When R = 2, the 2n-th moments coincide with the Catalan numbers (1, 2, 5, 14, ...); excess kurtosis = -1
• It is equivalent to a shifted/scaled beta distribution with parameters alpha = beta = 3/2
• In free probability theory the semicircle plays the role the normal distribution plays in classical probability: a distribution whose free cumulants of degree > 2 are all zero
• Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner semicircle distribution

Connections: Eugene Wigner · Random Matrix Theory · Probability Distributions · Free Probability

Source: http://en.wikipedia.org/wiki/Wigner_semicircle_distribution