Science China Mathematics

, Volume 62, Issue 5, pp 921–934

Multidimensional compound Poisson distributions in free probability

• Guimei An
• Mingchu Gao
Articles

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 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.

Keywords

free probability multidimensional free Poisson distributions multidimensional free infinitely divisible distributions

46L54

Notes

Acknowledgements

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.

References

1. 1.
Barndorff-Nielson O E, Thorjornsen S. Classical and free infinite divisibility and Lévy processes. In: Lecture Note in Mathematics, vol. 1866. Berlin-Heidelberg: Springer-Verlarg, 2006, 33–160Google Scholar
2. 2.
Bercovici H, Pata V. Stable laws and domains of attraction in free probabilty theory. Ann of Math (2), 1999, 149: 1023–1060
3. 3.
Feller W. An Introduction to Probability Theory and Its Applications. Volume II, 2nd ed. New York: John Wiley and Sons, 1970Google Scholar
4. 4.
Gao M. Two-faced families of non-commutative random variables having bi-free infinitely divisible distributions. Internat J Math, 2016, 27: 1650037
5. 5.
Ge L, Hadwin D. Ultraproducts of C*-algebras. In: Operator Theory: Advances and Applications, vol. 127. Basel: Beikhäuser, 2001, 305–326Google Scholar
6. 6.
Glockner P, Schurmann M, Speicher R. Realization of free white noises. Arch Math, 1992, 58: 407–416
7. 7.
Gu Y, Huang H, Mingo J. An analogue of the Lévy-Hinchin formula for bi-free infinitely divisible distributions. Indiana Univ Math J, 2016, 65: 1795–1831
8. 8.
Kadison R, Ringrose J. Fundamentals of the Theory of Operator Algebras. Graduate Studies in Mathematics, vol. 16. Providence: Amer Math Soc, 1997Google Scholar
9. 9.
Nica A, Speicher R. Lectures on Combinatorics for Free Probability. Cambridge: Cambridge University Press, 2006
10. 10.
Sinclair A, Smith R. Finite Von Neumann Algebras and Masas. Cabmbridge: Cabmbridge University Press, 2008
11. 11.
Speicher R. A new example of 'independence' and 'white noise'. Probab Theory Related Fields, 1990, 84: 141–159
12. 12.
Speicher R. Combinatorial Theory for the Free Product with Amalgamation and Operator-Valued Free Probability Theory. Memoirs of the American Mathematical Society, vol. 132. Providence: Amer Math Soc, 1998Google Scholar
13. 13.
Voiculescu D. Symmetries of Some Reduced Free Product C*-Algebras. Lecture Notes in Mathematics, vol. 1132. Berlin-Heidelberg: Spring Verlag, 1985, 556–588Google Scholar
14. 14.
Voiculescu D. Limit laws for random matrices and free products. Invent Math, 1991, 104: 201–220
15. 15.
Voicilescu D. Free probability for pairs of faces. Comm Math Phys, 2014, 332: 955–980
16. 16.
Voiculescu D, Dykema K, Nica A. Free Random Variables. Providence: Amer Math Soc, 1992