Marchenko–Pastur distribution

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Plot of the Marchenko-Pastur distribution for various values of lambda

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 (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



The Marchenko–Pastur law also arises as the free Poisson law in free probability theory, having rate and jump size .

Cumulative distribution function[edit]

Using the same notation, cumulative distribution function reads

where and .

Some transforms of this law[edit]

The Cauchy transform (which is the negative of the Stieltjes transformation), when , is given by

This gives an -transform of:

Application to correlation matrices[edit]

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 (i.e. 252 trading days) 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.

See also[edit]


  • 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.
  • Marchenko, V. A.; Pastur, L. A. (1967). "Распределение собственных значений в некоторых ансамблях случайных матриц" [Distribution of eigenvalues for some sets of random matrices]. Mat. Sb. N.S. (in Russian). 72 (114:4): 507–536. Bibcode:1967SbMat...1..457M. doi:10.1070/SM1967v001n04ABEH001994. Link to free-access pdf of Russian version
  • Nica, A.; Speicher, R. (2006). Lectures on the Combinatorics of Free probability theory. Cambridge Univ. Press. pp. 204, 368. ISBN 0-521-85852-6. Link to free download Another free access site
  • Zhang, W.; Abreu, G.; Inamori, M.; Sanada, Y. (2011). "Spectrum sensing algorithms via finite random matrices". IEEE Transactions on Communications. 60 (1): 164–175. doi:10.1109/TCOMM.2011.112311.100721. S2CID 206642535.
  • Epps, Brenden; Krivitzky, Eric M. (2019). "Singular value decomposition of noisy data: mode corruption". Experiments in Fluids. 60 (8): 1–30. Bibcode:2019ExFl...60..121E. doi:10.1007/s00348-019-2761-y. S2CID 198436243.