Marchenko–Pastur distribution

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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} X X^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} +\frac{1}{\lambda} \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 \lambda and jump size \sigma.

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