|
|
|
Viewing Version
2
of
'infinitely divisible random variable'
|
[ view 'infinitely divisible random variable'
|
back to history
]
| Title of object: |
infinitely divisible random variable |
| Canonical Name: |
InfinitelyDivisibleRandomVariable |
| Type: |
Definition |
| Created on: |
2006-11-24 14:13:45 |
| Modified on: |
2006-11-24 21:01:25 |
| Classification: |
msc:60E07 |
| Defines: |
infinitely divisible distribution, infinitely divisible |
Preamble:
\usepackage{amssymb,amscd}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
\usepackage{xypic}
\usepackage{pst-plot}
\usepackage{psfrag}
% define commands here
|
Content:
A real random variable $X$ defined on a probability space $(\Omega, \mathcal{F}, P)$ is said to be \emph{infinitely divisible} if for any positive integer $n$, $X$ is identically distributed as the sum of $n$ iid random variables $X_1,\ldots,X_n$.
A distribution function is said to be \emph{infinitely divisible} if it is the distribution function of an infinitely divisible random variable.
\textbf{Remark}. Any stable random variable is infinitely divisible.
Some examples of infinitely divisible distribution functions, besides those that are stable, are the gamma distributions, negative binomial distributions, and compound Poisson distributions. |
|
|
|
|
|
