  We study the high-dimensional asymptotic regimes of correlated Wishart
matrices $d^{-1}\mathcal{Y}\mathcal{Y}^T$, where $\mathcal{Y}$ is a $n\times d$
Gaussian random matrix with correlated and non-stationary entries. We prove
that under different normalizations, two distinct regimes emerge as both $n$
and $d$ grow to infinity. The first regime is the one of central convergence,
where the law of the properly renormalized Wishart matrices becomes close in
Wasserstein distance to that of a Gaussian orthogonal ensemble matrix. In the
second regime, a non-central convergence happens, and the law of the normalized
Wishart matrices becomes close in Wasserstein distance to that of the so-called
Rosenblatt-Wishart matrix recently introduced by Nourdin and Zheng. We then
proceed to show that the convergences stated above also hold in a functional
setting, namely as weak convergence in $C([a,b];M_n(\mathbb{R}))$. As an
application of our main result (in the central convergence regime), we show
that it can be used to prove convergence in expectation of the empirical
spectral distributions of the Wishart matrices to the semicircular law. Our
findings complement and extend a rich collection of results on the study of the
fluctuations of Gaussian Wishart matrices, and we provide explicit examples
based on Gaussian entries given by normalized increments of a bi-fractional or
a sub-fractional Brownian motion.
