Marchenko–Pastur distribution

MarchenkoPastur distribution


In random matrix theory, the MarchenkoPastur distribution, or MarchenkoPastur 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 = X X^T \,

and let \lambda_1,\, \lambda_2, \,\dots,\, \lambda_M be the eigenvalues of YN (viewed as random numbers). 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 } 0\leq \lambda \leq 1\\
\nu(A),& \text{if } \lambda ><span class=1, \end{cases} " border="0">

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 MarchenkoPastur law also arises as the free Poisson law in free probability theory, having rate λ and jump size α.

See also

References


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