Stable laws and domains of attraction in free probability theory

• 成果类型: Article
• 署名作者: Bercovici, H; Pata, V
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/121080
• 发表日期: 1999
• 页码: 1023-1060
• 关键词: random-variables; noncommutative probability; free convolution; large numbers

摘要

In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite different. Our work subsumes all previously known instances of weak convergence of sums of free, identically distributed random variables. In particular, we determine the domains of attraction of stable distributions in the free theory. These freely stable distributions are studied in detail in the appendix, where their unimodality and duality properties are demonstrated.