Marchenko–Pastur distribution

In the mathematical theory of

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

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

counting the number of eigenvalues in the subset ${\displaystyle A}$ included in ${\displaystyle \mathbb {R} }$.

Theorem. ^{[citation needed]} 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

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

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

Moments

For each ${\displaystyle k\geq 1}$, its ${\displaystyle k}$-th moment is^[1]

$${\displaystyle \sum _{r=0}^{k-1}{\frac {\sigma ^{2k}}{r+1}}{\binom {k}{r}}{\binom {k-1}{r}}\lambda ^{r}={\frac {\sigma ^{2k}}{k}}\sum _{r=0}^{k-1}{\binom {k}{r}}{\binom {k}{r+1}}\lambda ^{r}}$$

Some transforms of this law

The

${\displaystyle s(z)={\frac {\sigma ^{2}(1-\lambda )-z+{\sqrt {(z-\sigma ^{2}(\lambda +1))^{2}-4\lambda \sigma ^{4}}}}{2\lambda z\sigma ^{2}}}}$

for complex numbers z of positive imaginary part, where the

The S-transform is given by[3]

${\displaystyle S(z)={\frac {1}{\sigma ^{2}(1+\lambda z)}}.}$

Application to correlation matrices

For the special case of correlation matrices, we know that ${\displaystyle \sigma ^{2}=1}$ and ${\displaystyle \lambda =m/n}$. This bounds the probability mass over the interval defined by

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

Since this distribution describes the spectrum of random matrices with mean 0, the eigenvalues of correlation matrices that fall inside of the aforementioned interval could be considered spurious or noise. For instance, obtaining a correlation matrix of 10 stock returns calculated over a 252 trading days period would render ${\displaystyle \lambda _{+}=\left(1+{\sqrt {\frac {10}{252}}}\right)^{2}\approx 1.43}$. Thus, out of 10 eigenvalues of said correlation matrix, only the values higher than 1.43 would be considered significantly different from random.

See also

• Wigner semicircle distribution
• Tracy–Widom distribution

References