Marchenko–Pastur distribution

In random matrix theory, 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 mathematicians Vladimir Marchenko and Leonid Pastur who proved this result in 1967.

If $X$ denotes a $M\times N$ random matrix whose entries are independent identically distributed random variables with mean 0 and variance $\sigma ^{2}<\infty$ , let

Y_{N}=N^{{-1}}XX^{T}\,

and let $\lambda _{1},\,\lambda _{2},\,\dots ,\,\lambda _{M}$ be the eigenvalues of $Y_{N}$ (viewed as random variables). Finally, consider the random measure

\mu _{M}(A)={\frac {1}{M}}\#\left\{\lambda _{j}\in A\right\},\quad A\subset {\mathbb {R}}.

Theorem. Assume that $M,\,N\,\to \,\infty$ so that the ratio $M/N\,\to \,\lambda \in (0,+\infty )$ . Then $\mu _{{M}}\,\to \,\mu$ (in weak* topology in distribution), where

\mu(A) =\begin{cases} (1-\frac{1}{\lambda}) \mathbf{1}_{0\in A} + \nu(A),& \text{if } \lambda >1\\ \nu(A),& \text{if } 0\leq \lambda \leq 1, \end{cases}

and

d\nu (x)={\frac {1}{2\pi \sigma ^{2}}}{\frac {{\sqrt {(\lambda _{{+}}-x)(x-\lambda _{{-}})}}}{\lambda x}}\,{\mathbf {1}}_{{[\lambda _{{-}},\lambda _{{+}}]}}\,dx

with

\lambda _{{\pm }}=\sigma ^{2}(1\pm {\sqrt {\lambda }})^{2}.\,

The Marchenko–Pastur law also arises as the free Poisson law in free probability theory, having rate $1/\lambda$ and jump size $\sigma ^{2}$ .

References

Götze, F and Tikhomirov, A. (2004) "Rate of convergence in probability to the Marchenko–Pastur law", Bernoulli, 10 (3), 503–548. doi:10.3150/bj/1089206408
Marchenko,V. A., Pastur, L. A. (1967) "Distribution of eigenvalues for some sets of random matrices", Mat. Sb. (N.S.), 72(114):4, 507–536 doi:10.1070/SM1967v001n04ABEH001994 Link to free-access pdf of Russian version
Nica, A.; Speicher, R. (2006) Lectures on the Combinatorics of Free probability theory, Cambridge Univ. Press ISBN 0-521-85852-6 (pp. 204, 368). Link to free download Another free access site

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Marchenko–Pastur distribution

See also

References