Limit theorems in infinitesimal non-commutative probability spaces

CUI Han-zhe; XUE Yi-feng

Issue 3

May 2012

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CUI Han-zhe, XUE Yi-feng. Limit theorems in infinitesimal non-commutative probability spaces[J]. Journal of East China Normal University (Natural Sciences), 2012, (3): 56-60,96.

Citation:

CUI Han-zhe, XUE Yi-feng. Limit theorems in infinitesimal non-commutative probability spaces[J]. Journal of East China Normal University (Natural Sciences), 2012, (3): 56-60,96.

CUI Han-zhe, XUE Yi-feng. Limit theorems in infinitesimal non-commutative probability spaces[J]. Journal of East China Normal University (Natural Sciences), 2012, (3): 56-60,96.

Citation:

CUI Han-zhe, XUE Yi-feng. Limit theorems in infinitesimal non-commutative probability spaces[J]. Journal of East China Normal University (Natural Sciences), 2012, (3): 56-60,96.

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Limit theorems in infinitesimal non-commutative probability spaces

CUI Han-zhe^1,2,
XUE Yi-feng¹

1.
Department of Mathematics, East China Normal University, Shanghai 200241, China;
2.
Department of Mathematics and Physics, Shanghai Dianji University, Shanghai 200240, China

Received Date: 2011-07-01
Rev Recd Date: 2011-10-01
Publish Date: 2012-05-25

Abstract

Abstract

This paper proved the central limit theorems in infinitesimal non-commutative probability spaces using $R$ transform under both one-dimension and multi-dimension cases. Also Poisson limit theorem was proved using the moments-cumulants formula. Lastly, the relation between the standard semicircular elements and free poisson elements was figured out by combinatorial analysis.
- infinitesimal non-commutative probability spaces,
- central limit theorems,
- Poisson limit theorem

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References(1)

References

[1]

{1}NICA A, SPEICHER R. Lectures on the Combinatorics of Free Probability [M]. London Mathematical Society Lecture Note Series 335. Cambridge: Cambridge University Press, 2006.
{2}REINER V. Non-crossing partitions for classical reflection group[J]. Discrete Mathematics, 1997, 177: 195-222.
{3}BIANE P, GOODMAN F, NICA A. Non-crossing cumulants of type B[J]. Transactions of the American Mathematical Society, 2003, 355: 2263-2303.
{4}BELINSCHI S, SHLYAKHTENKO D. Free probability of type B: analytic aspects and applications [J/OL]. Preprint, arXiv: 0903.2721, 2009.
{5}FEVRIER M, NICA A. Infinitesimal non-crossing cumulants and free probability of Type B [J]. J Funct Anal, 2010, 258: 2983-2023.

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