A note on gaussian distributions in R^n

TL;DR

This paper explores the conditions under which non-Gaussian probability measures in R^n can have Gaussian marginals on certain subspaces, highlighting a distinction between finite and infinite families of subspaces.

Contribution

It provides examples of non-Gaussian measures with Gaussian marginals on finite sets of subspaces and proves the non-existence for infinite families.

Findings

Existence of non-Gaussian measures with Gaussian marginals on finite subspace sets

Non-existence of such measures when the family of subspaces is infinite

Clarification of the relationship between subspace family size and measure properties

Abstract

Given any finite set F of (n - 1)-dimensional subspaces of R^n we give examples of nongaussian probability measures in R^n whose marginal distribution in each subspace from F is gaussian. However, if F is an infinite family of such (n - 1)-dimensional subspaces then such a nongaussian probability measure in R^n does not exist.