On the Projections of Isotropic Distributions

Abstract

The class of distributions on $R^m, 1 \leq m < \infty$ which are the $m$-dimensional marginal distributions of orthogonally invariant distributions on $R^{m + n}$ is characterized. This result is then used to provide a partial answer to the following question: given a symmetric distribution on $R^1$ and an integer $n \geq 2$, under what conditions will there exist a random vector $X \in R^n$ such that $a'X$ has the given distribution (up to a positive scale factor) for all $a \neq 0, a \in R^m$.