WIGNER CHAOS AND THE FOURTH MOMENT

• 成果类型: Article
• 署名作者: Kemp, Todd; Nourdin, Ivan; Peccati, Giovanni; Speicher, Roland
• 署名单位: University of California System; University of California San Diego; Universite de Lorraine; University of Luxembourg; Saarland University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP657
• 发表日期: 2012
• 页码: 1577-1635
• 关键词: free stochastic-measures; central limit-theorems; free diffusions

摘要

We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2, the fourth moment of the semicircular law. This extends to the free probabilistic, setting some recent results by Nualart and Peccati on characterizations of central limit theorems in a fixed order of Gaussian Wiener chaos. Our proof is combinatorial, analyzing the relevant noncrossing partitions that control the moments of the integrals. We can also use these techniques to distinguish the first order of chaos from all others in terms of distributions; we then use tools from the free Malliavin calculus to give quantitative bounds on a distance between different orders of chaos. When applied to highly symmetric kernels, our results yield a new transfer principle, connecting central limit theorems in free Wigner chaos to those in Gaussian Wiener chaos. We use this to prove a new free version of an important classical theorem, the Breuer-Major theorem.