Marchenko–Pastur distribution

Using the same notation, cumulative distribution function reads

${\displaystyle F_{\lambda }(x)={\begin{cases}{\frac {\lambda -1}{\lambda }}\mathbf {1} _{x\in [0,\lambda _{-})}+\left({\frac {\lambda -1}{2\lambda }}+F(x)\right)\mathbf {1} _{x\in [\lambda _{-},\lambda _{+})}+\mathbf {1} _{x\in [\lambda _{+},\infty )},&{\text{if }}\lambda >1\\F(x)\mathbf {1} _{x\in [\lambda _{-},\lambda _{+})}+\mathbf {1} _{x\in [\lambda _{+},\infty )},&{\text{if }}0\leq \lambda \leq 1,\end{cases}}}$

where ${\textstyle F(x)={\frac {1}{2\pi \lambda }}\left(\pi \lambda +\sigma ^{-2}{\sqrt {(\lambda _{+}-x)(x-\lambda _{-})}}-(1+\lambda )\arctan {\frac {r(x)^{2}-1}{2r(x)}}+(1-\lambda )\arctan {\frac {\lambda _{-}r(x)^{2}-\lambda _{+}}{2\sigma ^{2}(1-\lambda )r(x)}}\right)}$ and ${\displaystyle r(x)={\sqrt {\frac {\lambda _{+}-x}{x-\lambda _{-}}}}}$.

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

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

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

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

Voiculescu's ${\displaystyle R}$-transform is given by

${\displaystyle R_{\mu }(z)={\frac {\sigma ^{2}}{1-\sigma ^{2}\lambda z}},}$

and the ${\displaystyle S}$-transform by

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

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.

• Wigner semicircle distribution
• Tracy–Widom distribution

• 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.
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