Uncertainty theory

2013-09-19T14:24:32Z

88.89.82.186: Reference 2 is now at 4th yet unpublished edition

In [[Random matrix|random matrix theory]], the '''Marchenko–Pastur distribution''', or '''Marchenko–Pastur law''', describes the [[asymptotic]] behavior of [[singular values]] of large rectangular [[random matrix|random matrices]]. The theorem is named after [[Ukraine|Ukrainian]] [[mathematicians]] [[Vladimir Marchenko]] and [[Leonid Pastur]] who proved this result in 1967.

If <math>X</math> denotes a <math>M\times N</math> random matrix whose entries are independent identically distributed random variables with mean 0 and variance <math>\sigma^2 < \infty</math>, let

: <math>Y_N = N^{-1} X X^T \, </math>

and let <math>\lambda_1,\, \lambda_2, \,\dots,\, \lambda_M</math> be the [[eigenvalue]]s of <math>Y_N</math> (viewed as [[random variable]]s). Finally, consider the random measure

: <math>\mu_M (A) = \frac{1}{M} \# \left\{ \lambda_j \in A \right\}, \quad A \subset \mathbb{R}. </math>

'''Theorem'''. Assume that <math>M,\,N \,\to\, \infty</math> so that the ratio <math>M/N \,\to\, \lambda \in (0, +\infty)</math>. Then <math>\mu_{M} \,\to\, \mu</math> (in [[Weak_topology#The_weak-.2A_topology|weak* topology]] in [[Convergence_in_distribution#Convergence_in_distribution|distribution]]), where
: <math>\mu(A) =\begin{cases} (1-\frac{1}{\lambda}) \mathbf{1}_{0\in A} + \nu(A),& \text{if } \lambda >1\\
\nu(A),& \text{if } 0\leq \lambda \leq 1,
\end{cases}
</math>
and
: <math>d\nu(x) = \frac{1}{2\pi \sigma^2 } \frac{\sqrt{(\lambda_{+} - x)(x - \lambda_{-})}}{\lambda x} \,\mathbf{1}_{[\lambda_{-}, \lambda_{+}]}\, dx</math>
with
: <math> \lambda_{\pm} = \sigma^2(1 \pm \sqrt{\lambda})^2. \, </math>

The Marchenko–Pastur law also arises as the [[free Poisson law]] in free probability theory, having rate <math>\lambda</math> and jump size <math>\alpha</math>.

==See also ==
* [[Wigner semicircle distribution]]

==References==
{{Reflist}}
*Götze, F and 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) "Distribution of eigenvalues for some sets of random matrices", ''Mat. Sb. (N.S.)'', 72(114):4, 507–536 {{doi|10.1070/SM1967v001n04ABEH001994}} [http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=4101&option_lang=eng Link to free-access pdf of Russian version]
*Nica, A.; [[Roland Speicher|Speicher, R.]] (2006) ''Lectures on the Combinatorics of Free probability theory'', Cambridge Univ. Press ISBN 0-521-85852-6 (pp. 204, 368). [http://www.ebooksdownloadfree.com/Science-Technology/Lectures-on-the-Combinatorics-of-Free-Probability-BI9525.html Link to free download] [http://www.google.com/search?tbs=bks:1&q=isbn:0521858526 Another free access site]
{{ProbDistributions|continuous-semi-infinite}}
{{DEFAULTSORT:Marchenko-Pastur distribution}}
[[Category:Probability distributions]]
[[Category:Random matrices]]

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Uncertainty theory