Wigner semicircle distribution

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

${\displaystyle E(X^{2n})=\left({R \over 2}\right)^{2n}C_{n}\,}$

where X is any random variable with this distribution and C_n is the nth Catalan number

${\displaystyle C_{n}={1 \over n+1}{2n \choose n},\,}$

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 ${\displaystyle x=R\cos(\theta )}$ into the defining equation for the moment generating function it can be seen that:

${\displaystyle M(t)={\frac {2}{\pi }}\int _{0}^{\pi }e^{Rt\cos(\theta )}\sin ^{2}(\theta )\,d\theta }$

which can be solved (see Abramowitz and Stegun §9.6.18) to yield:

${\displaystyle M(t)=2\,{\frac {I_{1}(Rt)}{Rt}}}$

where ${\displaystyle I_{1}(z)}$ is the modified Bessel function. Similarly, the characteristic function is given by:[1][2][3]

${\displaystyle \varphi (t)=2\,{\frac {J_{1}(Rt)}{Rt}}}$

where ${\displaystyle J_{1}(z)}$ is the Bessel function. (See Abramowitz and Stegun §9.1.20), noting that the corresponding integral involving ${\displaystyle \sin(Rt\cos(\theta ))}$ is zero.)

In the limit of ${\displaystyle R}$ approaching zero, the Wigner semicircle distribution becomes a Dirac delta function.

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.

The normalized N-sphere probability density function supported on the interval [−1, 1] of radius 1 centered at (0, 0):

${\displaystyle f_{n}(x;n)={(1-x^{2})^{(n-1)/2}\Gamma (1+n/2) \over {\sqrt {\pi }}\Gamma ((n+1)/2)}\,(n>=-1)}$,

for −1 ≤ x ≤ 1, and f(x) = 0 if |x| > 1.

Example. The joint distribution is

${\displaystyle \int _{-{\sqrt {1-y^{2}-x^{2}}}}^{+{\sqrt {1-y^{2}-x^{2}}}}\int _{-{\sqrt {1-x^{2}}}}^{+{\sqrt {1-x^{2}}}}\int _{0}^{1}f_{X,Y,Z}(x,y,z){{\sqrt {1-x^{2}-y^{2}-z^{2}}}^{(n)}}dxdydz=1;}$

${\displaystyle f_{X,Y,Z}(x,y,z)={\frac {3}{4\pi }}}$

Hence, the marginal PDF distribution is [4]

${\displaystyle f_{X}(x;n)={(1-x^{2})^{(n-1)/2)}\Gamma (1+n/2) \over \ {\sqrt {\pi }}\Gamma ((n+1)/2)}\,;}$ such that R=1

The cumulative distribution function (CDF) is

${\displaystyle F_{X}(x)={2x\Gamma (1+n/2)_{2}F_{1}(1/2,(1-n)/2;3/2;x^{2}) \over \ {\sqrt {\pi }}\Gamma ((n+1)/2)}\,;}$ such that R=1 and n >= -1

The characteristic function (CF) of the PDF is related to the beta distribution as shown below

${\displaystyle CF(t;n)={_{1}F_{1}(n/2,;n;jt/2)}\,\urcorner (\alpha =\beta =n/2);}$

In terms of Bessel functions this is

${\displaystyle CF(t;n)={\Gamma (n/2+1)J_{n/2}(t)/(t/2)^{(n/2)}}\,\urcorner (n>=-1);}$

Raw moments of the PDF are

${\displaystyle \mu '_{N}(n)=\int _{-1}^{+1}x^{N}f_{X}(x;n)dx={(1+(-1)^{N})\Gamma (1+n/2) \over \ {2{\sqrt {\pi }}}\Gamma ((2+n+N)/2)};}$

Central moments are

${\displaystyle \mu _{0}(x)=1}$

${\displaystyle \mu _{1}(n)=\mu _{1}'(n)}$

${\displaystyle \mu _{2}(n)=\mu _{2}'(n)-\mu _{1}'^{2}(n)}$

${\displaystyle \mu _{3}(n)=2\mu _{1}'^{3}(n)-3\mu _{1}'(n)\mu _{2}'(n)+\mu _{3}'(n)}$

${\displaystyle \mu _{4}(n)=-3\mu _{1}'^{4}(n)+6\mu _{1}'^{2}(n)\mu _{2}'(n)-4\mu '_{1}(n)\mu '_{3}(n)+\mu '_{4}(n)}$

The corresponding probability moments (mean, variance, skew, kurtosis and excess-kurtosis) are:

${\displaystyle \mu (x)=\mu _{1}'(x)=0}$

${\displaystyle \sigma ^{2}(n)=\mu _{2}'(n)-\mu ^{2}(n)=1/(2+n)}$

${\displaystyle \gamma _{1}(n)=\mu _{3}/\mu _{2}^{3/2}=0}$

${\displaystyle \beta _{2}(n)=\mu _{4}/\mu _{2}^{2}=3(2+n)/(4+n)}$

${\displaystyle \gamma _{2}(n)=\mu _{4}/\mu _{2}^{2}-3=-6/(4+n)}$

Raw moments of the characteristic function are:

${\displaystyle \mu '_{N}(n)=\mu '_{N;E}(n)+\mu '_{N;O}(n)=\int _{-1}^{+1}cos^{N}(xt)f_{X}(x;n)dx+\int _{-1}^{+1}sin^{N}(xt)f_{X}(x;n)dx;}$

For an even distribution the moments are [5]

${\displaystyle \mu _{1}'(t;n:E)=CF(t;n)}$

${\displaystyle \mu _{1}'(t;n:O)=0}$

${\displaystyle \mu _{1}'(t;n)=CF(t;n)}$

${\displaystyle \mu _{2}'(t;n:E)=1/2(1+CF(2t;n))}$

${\displaystyle \mu _{2}'(t;n:O)=1/2(1-CF(2t;n))}$

${\displaystyle \mu '_{2}(t;n)=1}$

${\displaystyle \mu _{3}'(t;n:E)=(CF(3t)+3CF(t;n))/4}$

${\displaystyle \mu _{3}'(t;n:O)=0}$

${\displaystyle \mu _{3}'(t;n)=(CF(3t;n)+3CF(t;n))/4}$

${\displaystyle \mu _{4}'(t;n:E)=(3+4CF(2t;n)+CF(4t;n))/8}$

${\displaystyle \mu _{4}'(t;n:O)=(3-4CF(2t;n)+CF(4t;n))/8}$

${\displaystyle \mu _{4}'(t;n)=(3+CF(4t;n))/4}$

Hence, the moments of the CF (provided N=1) are

${\displaystyle \mu (t;n)=\mu _{1}'(t)=CF(t;n)=_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})}$

${\displaystyle \sigma ^{2}(t;n)=1-|CF(t;n)|^{2}=1-|_{0}F_{1}({2+n \over 2},-t^{2}/4)|^{2}}$

${\displaystyle \gamma _{1}(n)={\mu _{3} \over \mu _{2}^{3/2}}={_{0}F_{1}({2+n \over 2},-9{t^{2} \over 4})-_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})+8|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})|^{3} \over 4(1-|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}))^{2}|^{(3/2)}}}$

${\displaystyle \beta _{2}(n)={\mu _{4} \over \mu _{2}^{2}}={3+_{0}F_{1}({2+n \over 2},-4t^{2})-(4_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})(_{0}F_{1}({2+n \over 2},-9{t^{2} \over 4}))+3_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})(-1+|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}|^{2})) \over 4(-1+|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}))^{2}|^{2}}}$

${\displaystyle \gamma _{2}(n)=\mu _{4}/\mu _{2}^{2}-3={-9+_{0}F_{1}({2+n \over 2},-4t^{2})-(4_{0}F_{1}({2+n \over 2},-t^{2}/4)(_{0}F_{1}({2+n \over 2},-9{t^{2} \over 4}))-9_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})+6|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}|^{3}) \over 4(-1+|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}))^{2}|^{2}}}$

Skew and Kurtosis can also be simplified in terms of Bessel functions.

The entropy is calculated as

${\displaystyle H_{N}(n)=\int _{-1}^{+1}f_{X}(x;n)\ln(f_{X}(x;n))dx}$

The first 5 moments (n=-1 to 3), such that R=1 are

${\displaystyle \ -\ln(2/\pi );n=-1}$

${\displaystyle \ -\ln(2);n=0}$

${\displaystyle \ -1/2+\ln(\pi );n=1}$

${\displaystyle \ 5/3-\ln(3);n=2}$

${\displaystyle \ -7/4-\ln(1/3\pi );n=3}$

The marginal PDF distribution with odd symmetry is [4]

${\displaystyle f{_{X}}(x;n)={(1-x^{2})^{(n-1)/2)}\Gamma (1+n/2) \over \ {\sqrt {\pi }}\Gamma ((n+1)/2)}\operatorname {sgn}(x)\,;}$ such that R=1

Hence, the CF is expressed in terms of Struve functions

${\displaystyle CF(t;n)={\Gamma (n/2+1)H_{n/2}(t)/(t/2)^{(n/2)}}\,\urcorner (n>=-1);}$

"The Struve function arises in the problem of the rigid-piston radiator mounted in an infinite baffle, which has radiation impedance given by" [6]

${\displaystyle Z={\rho c\pi a^{2}[R_{1}(2ka)-iX_{1}(2ka)],}}$

${\displaystyle R_{1}={1-{2J_{1}(x) \over 2x},}}$

${\displaystyle X_{1}={{2H_{1}(x) \over x},}}$

The normalized received signal strength is defined as

${\displaystyle |R|={{1 \over N}|}\sum _{k=1}^{N}\exp[ix_{n}t]|}$

and using standard quadrature terms

${\displaystyle x={1 \over N}\sum _{k=1}^{N}\cos(x_{n}t)}$

${\displaystyle y={1 \over N}\sum _{k=1}^{N}\sin(x_{n}t)}$

Hence, for an even distribution we expand the NRSS, such that x = 1 and y = 0, obtaining

${\displaystyle {\sqrt {x^{2}+y^{2}}}=x+{3 \over 2}y^{2}-{3 \over 2}xy^{2}+{1 \over 2}x^{2}y^{2}+O(y^{3})+O(y^{3})(x-1)+O(y^{3})(x-1)^{2}+O(x-1)^{3}}$

The expanded form of the Characteristic function of the received signal strength becomes [7]

${\displaystyle E[x]={1 \over N}CF(t;n)}$

${\displaystyle E[y^{2}]={1 \over 2N}(1-CF(2t;n))}$

${\displaystyle E[x^{2}]={1 \over 2N}(1+CF(2t;n))}$

${\displaystyle E[xy^{2}]={t^{2} \over 3N^{2}}CF(t;n)^{3}+({N-1 \over 2N^{2}})(1-tCF(2t;n))CF(t;n)}$

${\displaystyle E[x^{2}y^{2}]={1 \over 8N^{3}}(1-CF(4t;n))+({N-1 \over 4N^{3}})(1-CF(2t;n)^{2})+({N-1 \over 3N^{3}})t^{2}CF(t;n)^{4}+({(N-1)(N-2) \over N^{3}})CF(t;n)^{2}(1-CF(2t;n))}$

• Wigner surmise
• The Wigner semicircle distribution is the limit of the Kesten–McKay distributions, as the parameter d tends to infinity.
• In number-theoretic literature, the Wigner distribution is sometimes called the Sato–Tate distribution. See Sato–Tate conjecture.
• Marchenko–Pastur distribution or Free Poisson distribution

• Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

• Eric W. Weisstein et al., Wigner's semicircle