THE MIDDLE-SCALE ASYMPTOTICS OF WISHART MATRICES

• 成果类型: Article
• 署名作者: Chetelat, Didier; Wells, Martin T.
• 署名单位: Universite de Montreal; Polytechnique Montreal; Cornell University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1760
• 发表日期: 2019
• 页码: 2639-2670
• 关键词: MOMENTS

摘要

We study the behavior of a real p-dimensional Wishart random matrix with n degrees of freedom when n, p -> infinity but p/n -> 0. We establish the existence of phase transitions when p grows at the order n((K+1)/(K+3)) for every K is an element of N, and derive expressions for approximating densities between every two phase transitions. To do this, we make use of a novel tool we call the F-conjugate of an absolutely continuous distribution, which is obtained from the Fourier transform of the square root of its density. In the case of the normalized Wishart distribution, this represents an extension of the t-distribution to the space of real symmetric matrices.