Marchenko–Pastur distribution
Introduction Plot of the Marchenko-Pastur distribution for various values of lambdaIn 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 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  (viewed as random variables). Finally, consider the random measure: counting the number of eigenvalues in the subset  included in . '''Theorem'''. Assume that  so that the ratio . Then  (in weak* topology in distribution), where: and: with: The Marchenko–Pastur law also arises as the free Poisson law in free probability theory, having rate  and jump size .
Cumulative distribution function Using the same notation, cumulative distribution function reads : whereand .
Moments For each , its -th moment is
Some transforms of this law The Cauchy transform (which is the negative of the Stieltjes transformation) is given by: Voiculescu's -transform is given by: and the -transform by:
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.
See also * Wigner semicircle distribution* Tracy–Widom distribution
References * *   Link to free-access pdf of Russian version*   Link to free download  Another free access site* *