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

$Y_N = N^{-1} X X^T \,$

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

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

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 $\sigma^2$.

See also

• Wigner semicircle distribution

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