Wigner semicircle distribution

Wigner semicircle

Probability density function
Cumulative distribution function
Parameters	$R>0\!$ radius (real)
Support	$x\in [-R;+R]\!$
PDF	${\frac {2}{\pi R^{2}}}\,{\sqrt {R^{2}-x^{2}}}\!$
CDF	${\frac {1}{2}}+{\frac {x{\sqrt {R^{2}-x^{2}}}}{\pi R^{2}}}+{\frac {\arcsin \!\left({\frac {x}{R}}\right)}{\pi }}\!$ for $-R\leq x\leq R$
Mean	$0\,$
Median	$0\,$
Mode	$0\,$
Variance	${\frac {R^{2}}{4}}\!$
Skewness	$0\,$
Excess kurtosis	$-1\,$
Entropy	$\ln(\pi R)-{\frac {1}{2}}\,$
MGF	$2\,{\frac {I_{1}(R\,t)}{R\,t}}$
CF	$2\,{\frac {J_{1}(R\,t)}{R\,t}}$

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):

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

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

${\frac {1}{n+1}}\left({R \over 2}\right)^{2n}{2n \choose n}\,$

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

$s(z)=-{\frac {2}{R^{2}}}(z-{\sqrt {z^{2}-R^{2}}})$

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:

$\varphi (t)=e^{-iRt}\varphi _{Y}(2Rt)=e^{-iRt}{}_{1}F_{1}\left({\frac {3}{2}};3;2iRt\right)={\frac {2J_{1}(Rt)}{Rt}},$

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

$M(t)=e^{-Rt}M_{Y}(2Rt)=e^{-Rt}{}_{1}F_{1}\left({\frac {3}{2}};3;2Rt\right)={\frac {2I_{1}(Rt)}{Rt}}$

where I₁ 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.

Related distributions

Wigner (spherical) parabolic distribution

Wigner parabolic

Parameters	$R>0\!$ radius (real)
Support	$x\in [-R;+R]\!$
PDF	${\frac {3}{4R^{3}}}\,(R^{2}-x^{2})$
CDF	${\frac {1}{4R^{3}}}\,(2R-x)\,(R+x)^{2}$
MGF	$3\,{\frac {i_{1}(R\,t)}{R\,t}}$
CF	$3\,{\frac {j_{1}(R\,t)}{R\,t}}$

The parabolic probability distribution supported on the interval [−R, R] of radius R centered at (0, 0):

$f(x)={3 \over \ 4R^{3}}{(R^{2}-x^{2})}\,$

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

Example. The joint distribution is

$\int _{0}^{\pi }\int _{0}^{+2\pi }\int _{0}^{R}f_{X,Y,Z}(x,y,z)R^{2}\,dr\sin(\theta )\,d\theta \,d\phi =1;$

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

Hence, the marginal PDF of the spherical (parametric) distribution is:

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

$f_{X}(x)=\int _{-{\sqrt {1-x^{2}}}}^{+{\sqrt {1-x^{2}}}}2{\sqrt {1-y^{2}-x^{2}}}\,dy\,;$

$f_{X}(x)={3 \over \ 4}{(1-x^{2})}\,;$ 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):

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

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

Example. The joint distribution is

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

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

Hence, the marginal PDF distribution is

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

The cumulative distribution function (CDF) is

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

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

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

In terms of Bessel functions this is

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

Raw moments of the PDF are

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

Central moments are

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

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

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

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

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

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

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

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

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

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

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

Raw moments of the characteristic function are:

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

For an even distribution the moments are

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The entropy is calculated as

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

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

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

$\ -\ln(2);n=0$

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

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

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

N-sphere Wigner distribution with odd symmetry applied

The marginal PDF distribution with odd symmetry is

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

Hence, the CF is expressed in terms of Struve functions

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

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

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

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

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

Example (Normalized Received Signal Strength): quadrature terms

The normalized received signal strength is defined as

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

and using standard quadrature terms

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

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

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

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

The expanded form of the Characteristic function of the received signal strength becomes

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

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

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

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

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

See also

• 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

References

• Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer (2010). An introduction to random matrices. Cambridge Studies in Advanced Mathematics. Vol. 118. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511801334. ISBN 978-0-521-19452-5. MR 2670897. Zbl 1184.15023.
• Bai, Zhidong; Silverstein, Jack W. (2010). Spectral analysis of large dimensional random matrices. Springer Series in Statistics (Second edition of 2006 original ed.). New York: Springer. doi:10.1007/978-1-4419-0661-8. ISBN 978-1-4419-0660-1. MR 2567175. Zbl 1301.60002.
• Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1995). Continuous univariate distributions. Volume 2. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (Second edition of 1970 original ed.). New York: John Wiley & Sons, Inc. ISBN 0-471-58494-0. MR 1326603. Zbl 0821.62001.
• Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010). NIST handbook of mathematical functions. Cambridge: Cambridge University Press. ISBN 978-0-521-14063-8. MR 2723248. Zbl 1198.00002.
• Wigner, Eugene P. (1955). "Characteristic vectors of bordered matrices with infinite dimensions". Annals of Mathematics. Second Series. 62 (3): 548–564. doi:10.2307/1970079. MR 0077805. Zbl 0067.08403.

External links

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