The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on [−R, R] whose probability density function f is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0):
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 Wigner distribution also coincides with a scaled beta distribution. That is, if Y is a beta-distributed random variable with parameters α = β = 3/2, then the random variable X = 2RY – R exhibits a Wigner semicircle distribution with radius R.
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
The Chebyshev polynomials of the third kind are orthogonal polynomials with respect to the Wigner semicircle distribution.
For positive integers n, the 2n-th moment of this distribution is
where X is any random variable with this distribution and Cn is the nth Catalan number
so that the moments are the Catalan numbers if R = 2. (Because of symmetry, all of the odd-order moments are zero.)
Making the substitution into the defining equation for the moment generating function it can be seen that:
which can be solved (see Abramowitz and Stegun §9.6.18) to yield:
where is the modified Bessel function. Similarly, the characteristic function is given by:
where is the Bessel function. (See Abramowitz and Stegun §9.1.20), noting that the corresponding integral involving is zero.)
In the limit of approaching zero, the Wigner semicircle distribution becomes a Dirac delta function.
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.
Related distributions
Wigner (spherical) parabolic distribution
The parabolic probability distribution supported on the interval [−R, R] of radius R centered at (0, 0):
for −R ≤ x ≤ R, and f(x) = 0 if |x| > R.
Example. The joint distribution is
Hence, the marginal PDF of the spherical (parametric) distribution is:
such that R=1
The characteristic function of a spherical distribution becomes the pattern multiplication of the expected values of the distributions in X, Y and Z.
The parabolic Wigner distribution is also considered the monopole moment of the hydrogen like atomic orbitals.
Wigner n-sphere distribution
The normalized N-sphere probability density function supported on the interval [−1, 1] of radius 1 centered at (0, 0):
,
for −1 ≤ x ≤ 1, and f(x) = 0 if |x| > 1.
Example. The joint distribution is
Hence, the marginal PDF distribution is
such that R=1
The cumulative distribution function (CDF) is
such that R=1 and n >= -1
The characteristic function (CF) of the PDF is related to the beta distribution as shown below
In terms of Bessel functions this is
Raw moments of the PDF are
Central moments are
The corresponding probability moments (mean, variance, skew, kurtosis and excess-kurtosis) are:
Raw moments of the characteristic function are:
For an even distribution the moments are
Hence, the moments of the CF (provided N=1) are
Skew and Kurtosis can also be simplified in terms of Bessel functions.
The entropy is calculated as
The first 5 moments (n=-1 to 3), such that R=1 are
N-sphere Wigner distribution with odd symmetry applied
The marginal PDF distribution with odd symmetry is
such that R=1
Hence, the CF is expressed in terms of Struve functions
"The Struve function arises in the problem of the rigid-piston radiator mounted in an infinite baffle, which has radiation impedance given by"