InfinitelyDivisibleRandomVariable 2006-11-24 14:13:45 2009-02-25 19:18:25 Definition $n$-decomposable $n$-divisible infinitely divisible distribution infinitely divisible decomposable random variable \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 Let $n$ be a positive integer. A real random variable $X$ defined on a probability space $(\Omega, \mathcal{F}, P)$ is said to be \begin{enumerate} \item \emph{$n$-decomposable} if there exist $n$ independent random variables $X_1,\ldots,X_n$ such that $X$ is identically distributed as the sum $X_1+\cdots+X_n$. A $2$-decomposable random variable is also called a \emph{decomposable random variable}; \item \emph{$n$-divisible} if $X$ is $n$-decomposable and the $X_i$'s can be chosen so that they are identically distributed; \item \emph{infinitely divisible} if $X$ is $n$-divisible for every positive integer $n$. In other words, $X$ can be written as the sum of $n$ iid random variables for any $n$. \end{enumerate} 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.