TY - JOUR

T1 - On the density of the sum of two independent Student t-random vectors

AU - Berg, Christian

AU - Vignat, Christophe

PY - 2010

Y1 - 2010

N2 - In this paper, we find an expression for the density of the sum of two independent d-dimensionalStudent t-random vectors X and Y with arbitrary degrees of freedom. As abyproduct we also obtain an expression for the density of the sum N+X, where N is normaland X is an independent Student t-vector. In both cases the density is given as an infiniteseries $\sum_{n=0}^\infty c_nf_n$where f_n is a sequence of probability densities on R^d and c_n is a sequence of positivenumbers of sum 1, i.e. the distribution of a non-negative integer-valued random variableC, which turns out to be infinitely divisible for d=1 and d=2.  When d=1 and thedegrees of freedom of the Student variables are equal, we recover an old result of Ruben. 

AB - In this paper, we find an expression for the density of the sum of two independent d-dimensionalStudent t-random vectors X and Y with arbitrary degrees of freedom. As abyproduct we also obtain an expression for the density of the sum N+X, where N is normaland X is an independent Student t-vector. In both cases the density is given as an infiniteseries $\sum_{n=0}^\infty c_nf_n$where f_n is a sequence of probability densities on R^d and c_n is a sequence of positivenumbers of sum 1, i.e. the distribution of a non-negative integer-valued random variableC, which turns out to be infinitely divisible for d=1 and d=2.  When d=1 and thedegrees of freedom of the Student variables are equal, we recover an old result of Ruben. 

KW - Faculty of Science

KW - sandsynlighedsregning, Student t-fordelinger

KW - Student t-distribution, convolution, infinite divisibility

U2 - 10.1016/j.spl.2010.02.019

DO - 10.1016/j.spl.2010.02.019

M3 - Journal article

VL - 80

SP - 1043

EP - 1055

JO - Statistics & Probability Letters

JF - Statistics & Probability Letters

SN - 0167-7152

ER -