The purpose of this paper is to
prove a converse to a theorem of Eaton and Perlman on convolutions of G-decreasing
functions. Both their result and our converse concern a connection between the
theory of reflection groups and a class of probability inequalities that are of interest
to statisticians. The original theorem states that the Convolution Theorem is
satisfied by reflection subgroups of the orthogonal group. We show in this paper
that if G is a finite linear group that satisfies the Convolution Theorem,
then G is a reflection group. Furthermore, we show that if ρ : G↪GL(V ) is
a faithful representation that satisfies the Convolution Theorem, then ρ
is a direct sum of the canonical Coxeter representation of G and a trivial
representation.
