Wigner semicircle distribution

Summary

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 semicircle
Probability density function
Plot of the Wigner semicircle PDF
Cumulative distribution function
Plot of the Wigner semicircle CDF
Parameters radius (real)
Support
PDF
CDF
for
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF

for −RxR, and f(x) = 0 if |x| > R.

It is also a scaled beta distribution: if Y is beta-distributed with parameters α = β = 3/2, then X = 2RYR has the Wigner semicircle distribution.

The distribution arises as the limiting distribution of eigenvalues of many random symmetric matrices as the size of the matrix approaches infinity. The distribution of the spacing between eigenvalues is addressed by the similarly named Wigner surmise.

General propertiesEdit

The Chebyshev polynomials of the second 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:[1][2][3]

 

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 probabilityEdit

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 distributionsEdit

Wigner (spherical) parabolic distributionEdit

Wigner parabolic
Parameters   radius (real)
Support  
PDF  
CDF  
MGF  
CF  

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

 

for −RxR, and f(x) = 0 if |x| > R.

Example. The joint distribution is

 

 

Hence, the marginal PDF of the spherical (parametric) distribution is:[4]

 

 

  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 distributionEdit

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 [4]

  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 [5]

 

 

 

 

 

 

 

 

 

 

 

 

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 appliedEdit

The marginal PDF distribution with odd symmetry is [4]

  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" [6]

 

 

 

Example (Normalized Received Signal Strength): quadrature termsEdit

The normalized received signal strength is defined as

 

and using standard quadrature terms

 

 

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

 

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

 

 

 

 

 

See alsoEdit

ReferencesEdit

  1. ^ Buchanan, Kristopher; Flores, Carlos; Wheeland, Sara; Jensen, Jeffrey; Grayson, David; Huff, Gregory (2017). "Transmit beamforming for radar applications using circularly tapered random arrays". 2017 IEEE Radar Conference (Radar Conf). pp. 0112–0117. doi:10.1109/RADAR.2017.7944181. ISBN 978-1-4673-8823-8. S2CID 38429370.
  2. ^ Ryan, Buchanan (29 May 2014). "Theory and Applications of Aperiodic (Random) Phased Arrays". {{cite journal}}: Cite journal requires |journal= (help)
  3. ^ Overturf, Drew; Buchanan, Kristopher; Jensen, Jeffrey; Wheeland, Sara; Huff, Gregory (2017). "Investigation of beamforming patterns from volumetrically distributed phased arrays". MILCOM 2017 - 2017 IEEE Military Communications Conference (MILCOM). pp. 817–822. doi:10.1109/MILCOM.2017.8170756. ISBN 978-1-5386-0595-0. S2CID 11591305. https://ieeexplore.ieee.org/abstract/document/8170756/
  4. ^ a b c Buchanan, K.; Huff, G. H. (July 2011). "A comparison of geometrically bound random arrays in euclidean space". 2011 IEEE International Symposium on Antennas and Propagation (APSURSI): 2008–2011. doi:10.1109/APS.2011.5996900. ISBN 978-1-4244-9563-4. S2CID 10446533.
  5. ^ Thomas M. Cover (1963). "Antenna patern distribution from random array" (PDF) (MEMORANDUM RM-3502--PR). Santa Monica: The RAND Corporation. Archived (PDF) from the original on September 4, 2021.
  6. ^ W., Weisstein, Eric. "Struve Function". mathworld.wolfram.com. Retrieved 2017-07-28.
  7. ^ "Advanced Beamforming for Distributed and Multi-Beam Networks" (PDF).

External linksEdit