# 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 mathematicians Vladimir Marchenko and Leonid Pastur who proved this result in 1967.

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

${\displaystyle Y_{n}={\frac {1}{n}}XX^{T}}$

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

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

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

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

and

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

with

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

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

## References

• Götze, F.; 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). "Распределение собственных значений в некоторых ансамблях случайных матриц" [Distribution of eigenvalues for some sets of random matrices]. Mat. Sb. N.S. (in Russian). 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. pp. 204, 368. ISBN 0-521-85852-6. Link to free download Another free access site