Limit theorems in infinitesimal non-commutative probability spaces
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摘要: 在无穷小非交换概率空间中, 利用\,$R$\,变换证明了一维和多维情形下的中心极限定理, 利用矩函数和累积函数的关系证明了泊松极限定理, 并通过组合分析的方法给出了标准半单位圆元素与自由泊松元素之间的关系.}
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关键词:
- 无穷小非交换概率空间 /
- 中心极限定理 /
- 泊松极限定理
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. -
[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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