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.