Marchenko–Pastur distribution
Encyclopedia
In random matrix theory
, 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 numbers). Finally, consider the random measure
Theorem. Assume that
so that the ratio 
. Then 
(in weak* topology in distribution), where
Random matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable. Many important properties of physical systems can be represented mathematically as matrix problems...
, the Marchenko–Pastur distribution, or Marchenko–Pastur law, describes the asymptotic behavior of singular values of large rectangular random matrices
Random matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable. Many important properties of physical systems can be represented mathematically as matrix problems...
. The theorem is named after Ukrainian mathematicians Vladimir Marchenko
Vladimir Marchenko
Vladimir Marchenko is a Ukrainian mathematician who specializes in mathematical physics, in particular in the analysis of the Sturm–Liouville operators. He introduced one of the approaches to the inverse problem for Sturm–Liouville operators...
and Leonid Pastur
Leonid Pastur
Leonid Andreevich Pastur is a mathematical physicist and theoretical physicist, known in particular for contributions to random matrix theory, the spectral theory of random Schrödinger operators, statistical mechanics, and solid state physics...
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 numbers). Finally, consider the random measure
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 
.
-
