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 $p/n\rightarrow 0$. 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 G-transform of a distribution, which is closely related to the characteristic function. We also derive an extension of the $t$-distribution to the real symmetric matrices, which naturally appears as the conjugate distribution to the Wishart under a G-transformation, and show its empirical spectral distribution obeys a semicircle law when $p/n\rightarrow 0$. Finally, we discuss how the phase transitions of the Wishart distribution might originate from changes in rates of convergence of symmetric $t$ statistics.