On the density of the sum of two independent Student [formula omitted]-random vectors

Abstract

In this paper, we find an expression for the density of the sum of two independent d -dimensional Student t -random vectors X and Y with arbitrary degrees of freedom. As a byproduct we also obtain an expression for the density of the sum N + X , where N is normal and X is an independent Student t -vector. In both cases the density is given as an infinite series ∑ n = 0 ∞ c n f n where f n is a sequence of probability densities on R d and ( c n ) is a sequence of positive numbers of sum 1 , i.e. the distribution of a non-negative integer-valued random variable C , which turns out to be infinitely divisible for d = 1 and d = 2 . When d = 1 and the degrees of freedom of the Student variables are equal, we recover an old result of Ruben.