Multidimensional compound Poisson distributions in free probability

Abstract

Inspired by Speicher's multidimensional free central limit theorem and semicircle families, we prove an infinite dimensional compound Poisson limit theorem in free probability, and define infinite dimensional compound free Poisson distributions in a non-commutative probability space. Infinite dimensional free infinitely divisible distributions are defined and characterized in terms of their free cumulants. It is proved that for a sequence of random variables, the following three statements are equivalent: (1) the distribution of the sequence is multidimensional free infinitely divisible; (2) the sequence is the limit in distribution of a sequence of triangular trays of families of random variables; (3) the sequence has the same distribution as that of $$\left\{ {a_1^{\left( i \right)}:i = 1,2 \ldots } \right\}$$ of a multidimensional free Levy process $$\left\{ {\left\{ {a_t^{\left( i \right)}:i = 1,2 \ldots } \right\}:t \geqslant 0} \right\}$$ . Under certain technical assumption, this is the case if and only if the sequence is the limit in distribution of a sequence of sequences of random variables having multidimensional compound free Poisson distributions.