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

TL;DR

This paper derives an infinite series expression for the density of the sum of two independent Student t-random vectors in arbitrary dimensions, extending known results and exploring properties like infinite divisibility.

Contribution

It provides a novel infinite series representation for the sum of Student t-vectors with arbitrary degrees of freedom and dimensions, generalizing previous results.

Findings

Density expressed as an infinite series involving probability densities

Distribution of the sum is infinitely divisible for dimensions 1 and 2

Recovers classical results for equal degrees of freedom in one dimension

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.