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 X and Y with arbitrary degrees of freedom. As a byproduct we also obtain an expression for the density of the sum Ni+X, where N is normal and X is an independent Student t-vector. In both cases the density is given as an infinite series Sigma(infinity)(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. (C) 2010 Elsevier B.V. All rights reserved.
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Christian Berg, Christophe Vignat. On the density of the sum of two independent Student t-random vectors. Statistics and Probability Letters, Elsevier, 2010, 80 (13-14), pp.1043-1055. ⟨10.1016/j.spl.2010.02.019⟩. ⟨hal-00693454⟩

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