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.

Plot of the Marchenko-Pastur distribution for various values of lambda

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 ${\displaystyle \sigma ^{2}}$.

Some transforms of this law

The Cauchy transform (which is the negative of the Stieltjes transformation) is given by

${\displaystyle G_{\mu }(z)={\frac {z+\lambda -1-{\sqrt {(z-\lambda -1)^{2}-4y}}}{2\lambda z}}}$

This gives an ${\displaystyle R}$-transform of:

${\displaystyle R_{\mu }(z)={\frac {1}{1-\lambda z}}}$

Application to correlation matrices

When applied to correlation matrices ${\displaystyle \sigma ^{2}=1}$ and ${\displaystyle \lambda =m/n}$ which leads to the bounds

${\displaystyle \lambda _{\pm }=\left(1\pm {\sqrt {\frac {m}{n}}}\right)^{2}.}$

Hence, it is often assumed that eigenvalues of correlation matrices lower than ${\displaystyle \lambda _{+}}$ are by a chance, and the values higher than ${\displaystyle \lambda _{+}}$ are the significant common factors. For instance, obtaining a correlation matrix of a year long series (i.e. 252 trading days) of 10 stock returns, would render ${\displaystyle \lambda _{+}=\left(1+{\sqrt {\frac {10}{252}}}\right)^{2}\approx 1.43}$. Out of 10 eigen values of the correlation matrix only the values higher than 1.43 would be considered significant.

See also

• Wigner semicircle distribution
• Tracy–Widom distribution

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