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 \rho:G\hookrightarrow GL(V) is a faithful representation that satisfies the Convolution Theorem, then \rho is a direct sum of the canonical Coxeter representation of G and a trivial representation.