Title: 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.