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 ${\displaystyle r(x)={\sqrt {\frac {\lambda _{+}-x}{x-\lambda _{-}}}}}$.

The Cauchy transform (which is the negative of the Stieltjes transformation), when ${\displaystyle \sigma ^{2}=1}$, is given by

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

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

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

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.

• 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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