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 Soviet Ukrainian mathematicians Volodymyr Marchenko and Leonid Pastur who proved this result in 1967.
If denotes a random matrix whose entries are independent identically distributed random variables with mean 0 and variance , let
and let be the eigenvalues of . Finally, consider the random measure
counting the number of eigenvalues in the subset included in.
Theorem. Assume that so that the ratio. Then , where
and
with
The Marchenko–Pastur law also arises as the free Poisson law in free probability theory, having rate and jump size.

Singular value bounds in the large system limit

As the dimensions of a random matrix grow larger, the max/min singular values converge to
These are useful approximations of singular value bounds for large matrices. For matrices of finite size as are typically encountered, they are more what you'd call "guidelines" than actual rules.

Moments

For each, its -th moment is

Some transforms of this law

The Stieltjes transform is given by
for complex numbers of positive imaginary part, where the complex square root is also taken to have positive imaginary part. It satisfies the quadratic equationThe Stieltjes transform can be repackaged in the form of the R-transform, which is given by
The S-transform is given by
For the case of, the -transform is given by where satisfies the Marchenko-Pastur law.
where
For exact analysis of high dimensional regression in the proportional asymptotic regime, a convenient form is often which simplifies to
The following functions and, where satisfies the Marchenko-Pastur law, show up in the limiting Bias and Variance respectively, of ridge regression and other regularized linear regression problems. One can show that and.

Application to correlation matrices

For the special case of correlation matrices, we know that
and. This bounds the probability mass over the interval defined by
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. Thus, out of 10 eigenvalues of said correlation matrix, only the values higher than 1.43 would be considered significantly different from random.