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 Ukrainian mathematicians Vladimir 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
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.

Some transforms of this law

The Cauchy transform is given by
This gives an -transform of:

Application to correlation matrices

When applied to correlation matrices
and which leads to the bounds
Hence, it is often assumed that eigenvalues of correlation matrices lower than are by a chance, and the values higher than are the significant common factors. For instance, obtaining a correlation matrix of a year long series of 10 stock returns, would render. Out of 10 eigen values of the correlation matrix only the values higher than 1.43 would be considered significant.