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.
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
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.[1] As can be calculated using the residue theorem, the Stieltjes transform of the Wigner distribution is given by
for complex numbers z with positive imaginary part, where the complex square root is taken to have positive imaginary part.[2]
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.[3] In particular, it is direct to recover the characteristic function of the Wigner distribution from that of Y:
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.[4]
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.
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. ISBN978-1-4419-0660-1. 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.ISBN0-471-58494-0. MR 1326603. Zbl 0821.62001.
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.
wigner, semicircle, distribution, named, after, physicist, eugene, wigner, probability, distribution, whose, probability, density, function, scaled, semicircle, semi, ellipse, centered, wigner, semicircleprobability, density, functioncumulative, distribution, . 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 Wigner semicircleProbability density functionCumulative distribution functionParametersR gt 0 displaystyle R gt 0 radius real Supportx R R displaystyle x in R R PDF2pR2R2 x2 displaystyle frac 2 pi R 2 sqrt R 2 x 2 CDF12 xR2 x2pR2 arcsin xR p displaystyle frac 1 2 frac x sqrt R 2 x 2 pi R 2 frac arcsin left frac x R right pi for R x R displaystyle R leq x leq R Mean0 displaystyle 0 Median0 displaystyle 0 Mode0 displaystyle 0 VarianceR24 displaystyle frac R 2 4 Skewness0 displaystyle 0 Excess kurtosis 1 displaystyle 1 Entropyln pR 12 displaystyle ln pi R frac 1 2 MGF2I1 Rt Rt displaystyle 2 frac I 1 R t R t CF2J1 Rt Rt displaystyle 2 frac J 1 R t R t f x 2pR2R2 x2 displaystyle f x 2 over pi R 2 sqrt R 2 x 2 for R x 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 Contents 1 General properties 2 Relation to free probability 3 See also 4 References 5 External linksGeneral properties editBecause 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 1n 1 R2 2n 2nn displaystyle frac 1 n 1 left R over 2 right 2n 2n choose n nbsp 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 1 As can be calculated using the residue theorem the Stieltjes transform of the Wigner distribution is given by s z 2R2 z z2 R2 displaystyle s z frac 2 R 2 z sqrt z 2 R 2 nbsp for complex numbers z with positive imaginary part where the complex square root is taken to have positive imaginary part 2 The Wigner distribution coincides with a scaled and shifted beta distribution if Y is a beta distributed random variable with parameters a b 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 3 In particular it is direct to recover the characteristic function of the Wigner distribution from that of Y f t e iRtfY 2Rt e iRt1F1 32 3 2iRt 2J1 Rt Rt 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 nbsp 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 RtMY 2Rt e Rt1F1 32 3 2Rt 2I1 Rt Rt displaystyle M t e Rt M Y 2Rt e Rt 1 F 1 left frac 3 2 3 2Rt right frac 2I 1 Rt Rt nbsp 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 4 The Chebyshev polynomials of the third kind are orthogonal polynomials with respect to the Wigner semicircle distribution of radius 1 5 Relation to free probability editIn 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 See also editWigner 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 distributionReferences edit Anderson Guionnet amp Zeitouni 2010 Section 2 1 1 Bai amp Silverstein 2010 Section 2 1 1 Anderson Guionnet amp Zeitouni 2010 Section 2 4 1 Bai amp Silverstein 2010 Section 2 3 1 Johnson Kotz amp Balakrishnan 1995 Section 25 3 See identities 10 16 5 and 10 39 5 of Olver et al 2010 See Table 18 3 1 of Olver et al 2010 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 amp 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 editEric W Weisstein et al Wigner s semicircle Retrieved from https en wikipedia org w index php title Wigner semicircle distribution amp oldid 1209455224, wikipedia, wiki, book, books, library,