## The Annals of Statistics

### The middle-scale asymptotics of Wishart matrices

#### 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.

#### Article information

Source
Ann. Statist., Volume 47, Number 5 (2019), 2639-2670.

Dates
Revised: August 2018
First available in Project Euclid: 3 August 2019

https://projecteuclid.org/euclid.aos/1564797859

Digital Object Identifier
doi:10.1214/18-AOS1760

Mathematical Reviews number (MathSciNet)
MR3988768

#### Citation

Chételat, Didier; Wells, Martin T. The middle-scale asymptotics of Wishart matrices. Ann. Statist. 47 (2019), no. 5, 2639--2670. doi:10.1214/18-AOS1760. https://projecteuclid.org/euclid.aos/1564797859

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