On the density of the sum of two independent Student tt-random vectors

In this paper, we find an expression for the density of the sum of two independent dd-dimensional Student tt-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 tt-vector. In both cases the density is given as an infinite series ∑n=0∞cnfn where fnfn is a sequence of probability densities on RdRd and (cn)(cn) is a sequence of positive numbers of sum 11, i.e. the distribution of a non-negative integer-valued random variable CC, which turns out to be infinitely divisible for d=1d=1 and d=2d=2. When d=1d=1 and the degrees of freedom of the Student variables are equal, we recover an old result of Ruben.