Marchenko–Pastur distribution

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

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

The Stieltjes transform is given by

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

for complex numbers z of positive imaginary part, where the complex square root is also taken to have positive imaginary part.^[2] The Stieltjes transform can be repackaged in the form of the R-transform, which is given by^[3]

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

The S-transform is given by^[3]

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

For the case of ${\displaystyle \sigma =1}$ , the ${\displaystyle \eta }$ -transform ^[3] is given by ${\displaystyle \mathbb {E} {\frac {1}{1+\gamma X}}}$  where ${\displaystyle X}$  satisfies the Marchenko-Pastur law.

${\displaystyle \eta (\gamma )=1-{\frac {{\mathcal {F}}(\gamma ,\lambda )}{4\gamma \lambda }}}$ 

where ${\displaystyle {\mathcal {F}}(x,z)=\left({\sqrt {x(1+{\sqrt {z}})^{2}+1}}-{\sqrt {x(1-{\sqrt {z}})^{2}+1}}\right)^{2}}$ 

A convenient form is often ${\displaystyle \eta \left({\tfrac {1}{t}}\right)}$  which simplifies to

${\displaystyle \eta ({\tfrac {1}{t}})={\frac {-1+\lambda -t+{\sqrt {(1+t-\lambda )^{2}+4t\lambda }}}{2\lambda }}}$ 

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