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October 2019 The middle-scale asymptotics of Wishart matrices
Didier Chételat, Martin T. Wells
Ann. Statist. 47(5): 2639-2670 (October 2019). DOI: 10.1214/18-AOS1760

Abstract

We study the behavior of a real $p$-dimensional Wishart random matrix with $n$ degrees of freedom when $n,p\rightarrow\infty$ but $p/n\rightarrow0$. We establish the existence of phase transitions when $p$ grows at the order $n^{(K+1)/(K+3)}$ for every $K\in\mathbb{N}$, and derive expressions for approximating densities between every two phase transitions. To do this, we make use of a novel tool we call the $\mathcal{F}$-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.

Citation

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Didier Chételat. Martin T. Wells. "The middle-scale asymptotics of Wishart matrices." Ann. Statist. 47 (5) 2639 - 2670, October 2019. https://doi.org/10.1214/18-AOS1760

Information

Received: 1 July 2017; Revised: 1 August 2018; Published: October 2019
First available in Project Euclid: 3 August 2019

zbMATH: 07114924
MathSciNet: MR3988768
Digital Object Identifier: 10.1214/18-AOS1760

Subjects:
Primary: 60B10 , 60B20
Secondary: 60E10

Keywords: Covariance estimation , Gaussian Orthogonal Ensemble , high-dimensional asymptotics , Phase transitions , random matric theory

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 5 • October 2019
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