Free Poisson distribution について

Words near each other

Dictionary Lists

mini英和辞書

mini和英辞書

翻訳と辞書　辞書検索 [ 開発暫定版 ]

スポンサードリンク

Free Poisson distribution ：ウィキペディア英語版

Free Poisson distribution

In the mathematics of free probability theory, the free Poisson distribution is a counterpart of the Poisson distribution in conventional probability theory.
==Definition==
The free Poisson distribution〔Free Random Variables by D. Voiculescu, K. Dykema, A. Nica, CRM Monograph Series, American Mathematical Society, Providence RI, 1992〕 with jump size

\alpha

and rate

\lambda

arises in free probability theory as the limit of repeated free convolution
:

\left( \left(1-\frac\right)\delta_0 + \frac\delta_\alpha\right)^

as ''N'' → ∞.
In other words, let

X_N

be random variables so that

X_N

has value

\alpha

with probability

\frac

and value 0 with the remaining probability. Assume also that the family

X_1,X_2,\ldots

are freely independent. Then the limit as

N\to\infty

of the law of

X_1+\cdots +X_N

is given by the Free Poisson law with parameters

\lambda,\alpha

.
This definition is analogous to one of the ways in which the classical Poisson distribution is obtained from a (classical) Poisson process.
The measure associated to the free Poisson law is given by
:

\mu=\begin (1-\lambda) \delta_0 + \lambda \nu,& \text  0\leq \lambda \leq 1 \\\nu, & \text\lambda >1,\end

where
:

\nu  = \frac\sqrt \, dt

and has support

((1-\sqrt)^2,\alpha (1+\sqrt)^2)

.
This law also arises in random matrix theory as the Marchenko–Pastur law. Its free cumulants are all equal to

\lambda

.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
■ウィキペディアで「Free Poisson distribution」の詳細全文を読む

スポンサードリンク

翻訳と辞書 : 翻訳のためのインターネットリソース