On Infinitely Divisible Random Vectors

Abstract

A normally distributed random vector $X$ is well known to be representable by $A \cdot Y$ (in the sense of having identical distributions), where $A$ is a matrix of constants and $Y$ is a random vector whose component random variables are independent. A necessary and sufficient condition for any infinitely divisible random vector to be so representable is given. The limiting case is discussed as are connections with the multivariate Poisson distribution and stochastic processes.