Marchenko–Pastur distribution について

Words near each other

Dictionary Lists

mini英和辞書

翻訳と辞書　辞書検索 [ 開発暫定版 ]

スポンサードリンク

Marchenko–Pastur law ：ウィキペディア英語版

Marchenko–Pastur distribution

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

X

denotes a

M\times N

random matrix whose entries are independent identically distributed random variables with mean 0 and variance

\sigma^2 < \infty

, let
:

Y_N = N^ X X^T \,

and let

\lambda_1,\, \lambda_2, \,\dots,\, \lambda_M

be the eigenvalues of

Y_N

(viewed as random variables). Finally, consider the random measure
:

\mu_M (A) = \frac \# \left\, \quad A \subset \mathbb.

Theorem. Assume that

M,\,N \,\to\, \infty

so that the ratio

M/N \,\to\, \lambda \in (0, +\infty)

. Then

\mu_ \,\to\, \mu

\mu(A) =\begin (1-\frac) \mathbf_ + \nu(A),& \text \lambda >1\\\nu(A),& \text 0\leq \lambda \leq 1,\end

and
:

d\nu(x) = \frac \frac)}} \,\mathbf_,">\lambda_ )}\, dx

with
:

\lambda_ = \sigma^2(1 \pm \sqrt)^2. \,

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

1/\lambda

and jump size

\sigma^2

翻訳と辞書 : 翻訳のためのインターネットリソース