Wigner semicircle distribution

The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on [&minus;R, R] whose probability density function f is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0):


 * $$f(x)={2 \over \pi R^2}\sqrt{R^2-x^2\,}\, $$

for &minus;R &le; x &le; R, and f(x) = 0 if |x| &gt; R. The parameter R is commonly referred to as the "radius" parameter of the distribution.

The distribution arises as the limiting distribution of the eigenvalues of many random symmetric matrices, that is, as the dimensions of the random matrix approach infinity. The distribution of the spacing or gaps between eigenvalues is addressed by the similarly named Wigner surmise.

General properties
Because of symmetry, all of the odd-order moments of the Wigner distribution are zero. For positive integers $n$, the $2n$-th moment of this distribution is
 * $$\frac{1}{n+1}\left({R \over 2}\right)^{2n} {2n\choose n}\, $$

In the typical special case that $R = 2$, this sequence coincides with the Catalan numbers 1, 2, 5, 14, etc. In particular, the second moment is $R^{2}/4$ and the fourth moment is $R^{4}/8$, which shows that the excess kurtosis is $−1$. As can be calculated using the residue theorem, the Stieltjes transform of the Wigner distribution is given by
 * $$s(z)=-\frac{2}{R^2}(z-\sqrt{z^2-R^2})$$

for complex numbers $z$ with positive imaginary part, where the complex square root is taken to have positive imaginary part.

The Wigner distribution coincides with a scaled and shifted beta distribution: if $Y$ is a beta-distributed random variable with parameters $α = β = 3/2$, then the random variable $2RY – R$ exhibits a Wigner semicircle distribution with radius $R$. By this transformation it is direct to compute some statistical quantities for the Wigner distribution in terms of those for the beta distributions, which are better known. In particular, it is direct to recover the characteristic function of the Wigner distribution from that of $Y$:
 * $$\varphi(t)=e^{-iRt}\varphi_Y(2Rt)=e^{-iRt}{}_1F_1\left(\frac{3}{2}; 3; 2iRt\right)=\frac{2J_1(Rt)}{Rt},$$

where $_{1}F_{1}$ is the confluent hypergeometric function and $J_{1}$ is the Bessel function of the first kind. Likewise the moment generating function can be calculated as
 * $$M(t)=e^{-Rt}M_Y(2Rt)=e^{-Rt}{}_1F_1\left(\frac{3}{2}; 3; 2Rt\right)=\frac{2I_1(Rt)}{Rt}$$

where $I_{1}$ is the modified Bessel function of the first kind. The final equalities in both of the above lines are well-known identities relating the confluent hypergeometric function with the Bessel functions.

The Chebyshev polynomials of the third kind are orthogonal polynomials with respect to the Wigner semicircle distribution of radius $1$.

Relation to free probability
In free probability theory, the role of Wigner's semicircle distribution is analogous to that of the normal distribution in classical probability theory. Namely, in free probability theory, the role of cumulants is occupied by "free cumulants", whose relation to ordinary cumulants is simply that the role of the set of all partitions of a finite set in the theory of ordinary cumulants is replaced by the set of all noncrossing partitions of a finite set. Just as the cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is normal, so also, the free cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is Wigner's semicircle distribution.