Multidimensional compound Poisson distributions in free probability

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2019,（5）
• 页码范围: 921-934
• 页数: 14页
• 语种: 英文

摘要

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 {al(i)):i =1,2,...}of a multidimensional free Lévy process {{at(i):i =1,2,...}:t ≥ 0}.Under certain technical assumptions,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.