Science China Mathematics

, Volume 62, Issue 5, pp 921–934 | Cite as

Multidimensional compound Poisson distributions in free probability

  • Guimei AnEmail author
  • Mingchu Gao


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 Lévy 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.


free probability multidimensional free Poisson distributions multidimensional free infinitely divisible distributions 




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This work was supported by National Natural Science Foundation of China (Grant Nos. 11101220, 11271199 and 11671214), Visiting Scholar Project Funded (Grant No. 96172373) and the Fundamental Research Funds for the Central Universities. The authors are grateful for the references's comments and suggestions.


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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical Sciences and LPMCNankai UniversityTianjinChina
  2. 2.School of Mathematics and Information ScienceBaoji University of Arts and SciencesBaojiChina
  3. 3.Department of MathematicsLouisiana CollegePinevilleUSA

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