On the density of the sum of two independent Student t-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 $\mathbf{X}$ and $\mathbf{Y}$ with arbitrary degrees of freedom. As a byproduct we also obtain an expression for the density of the sum $\mathbf{N}+\mathbf{X}$, where $\mathbf{N}$ is normal and $\mathbf{X}$ is an independent Student $t-$vector. In both cases the density is given as an infinite series \[ \sum_{n=0}^{\infty} c_{n}f_{n} \] where $f_{n}$ is a sequence of probability densities on $\mathbb{R}^{d}$ 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.