The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution defined on the domain [−R, R] whose probability density function f is a scaled semicircle, i.e. a semi-ellipse, centered at (0, 0):
f ( x ) = 2 π R 2 R 2 − x 2 {\displaystyle f(x)={2 \over \pi R^{2}}{\sqrt {R^{2}-x^{2}\,}}\,}
for −R ≤ x ≤ R, and f(x) = 0 if |x| > 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
1 n + 1 ( R 2 ) 2 n ( 2 n n ) {\displaystyle {\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 R2⁄4 and the fourth moment is R4⁄8, 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 ) = − 2 R 2 ( z − z 2 − R 2 ) {\displaystyle 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 straightforward to directly compute some statistical quantities for the Wigner distribution in terms of those for the beta distributions, which are better known. The Chebyshev polynomials of the second kind are orthogonal polynomials with respect to the Wigner semicircle distribution of radius 1.
Characteristic function and moment generating function The characteristic function of the Wigner distribution can be determined from that of the beta-variate Y:
φ ( t ) = e − i R t φ Y ( 2 R t ) = e − i R t
1 F 1 ( 3 2 ; 3 ; 2 i R t ) = 2 J 1 ( R t ) R t , {\displaystyle \varphi (t)=e^{-iRt}\varphi _{Y}(2Rt)=e^{-iRt}{}_{1}F_{1}\left({\frac {3}{2}};3;2iRt\right)={\frac {2J_{1}(Rt)}{Rt}},}
where 1F1 is the confluent hypergeometric function and J1 is the Bessel function of the first kind. Likewise the moment generating function can be calculated as
M ( t ) = e − R t M Y ( 2 R t ) = e − R t
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