The Wigner semicircle distribution, named after the physicist
Eugene Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his co ...
, is the
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
on
minus;''R'', ''R''whose
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
''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''.
It is also a scaled
beta distribution
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
: if ''Y'' is beta-distributed with parameters α = β = 3/2, then ''X'' = 2''RY'' – ''R'' has the Wigner semicircle distribution.
The distribution arises as the limiting distribution of
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
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 properties
The
Chebyshev polynomials
The Chebyshev polynomials are two sequences of polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric ...
of the third kind are
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the class ...
with respect to the Wigner semicircle distribution.
For positive integers ''n'', the 2''n''-th
moment
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of this distribution is
:
where ''X'' is any random variable with this distribution and ''C''
''n'' is the ''n''th
Catalan number
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Ca ...
:
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 Stegu
§9.6.18)to yield:
:
where
is the modified
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrar ...
. Similarly, the characteristic function is given by:
:
where
is the Bessel function. (See Abramowitz and Stegu
§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
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
.
Relation to free probability
In
free probability theory, the role of Wigner's semicircle distribution is analogous to that of the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
in classical probability theory. Namely,
in free probability theory, the role of
cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s 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
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
are all zero
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
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
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
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
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, call ...
probability density function supported on the interval
��1, 1of 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
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
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"
Example (Normalized Received Signal Strength): quadrature terms
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
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
In the mathematical theory of random matrices, the Marchenko–Pastur distribution, or Marchenko–Pastur law, describes the asymptotic behavior of singular values of large rectangular random matrices. The theorem is named after Ukrainian mathem ...
or
Free Poisson distribution
References
* Milton Abramowitz and Irene A. Stegun, eds. ''
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.'' New York: Dover, 1972.
External links
*
Eric W. Weisstein
Eric Wolfgang Weisstein (born March 18, 1969) is an American mathematician and encyclopedist who created and maintains the encyclopedias ''MathWorld'' and ''ScienceWorld''. In addition, he is the author of the '' CRC Concise Encyclopedia of M ...
et al.
Wigner's semicircle
{{ProbDistributions, continuous-bounded
Continuous distributions