On the Projections of Isotropic Distributions
The Annals of Statistics · 1981 · Volume 9 · Issue 2
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 $a \neq 0, a \in R^m$.
Keywords
$n$-dimensional distributions, 60E05, 62E10, 62H05, orthogonally invariant distributions, scale mixtures of normals
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