  In this note, we study the Gaussian fluctuations for the Wishart matrices
$d^{-1}\mathcal{X}_{n, d}\mathcal{X}^{T}_{n, d}$, where $\mathcal{X}_{n, d}$ is
a $n\times d$ random matrix whose entries are jointly Gaussian and correlated
with row and column covariance functions given by $r$ and $s$ respectively such
that $r(0)=s(0)=1$. Under the assumptions $s\in \ell^{4/3}(\mathbb{Z})$ and
$\|r\|_{\ell^1(\mathbb{Z})}< \sqrt{6}/2$, we establish the $\sqrt{n^3/d}$
convergence rate for the Wasserstein distance between a normalization of
$d^{-1}\mathcal{X}_{n, d}\mathcal{X}^{T}_{n, d}$ and the corresponding Gaussian
ensemble. This rate is the same as the optimal one computed in
\cite{JL15,BG16,BDER16} for the total variation distance, in the particular
case where the Gaussian entries of $\mathcal{X}_{n, d}$ are independent.
Similarly, we obtain the $\sqrt{n^{2p-1}/d}$ convergence rate for the
Wasserstein distance in the setting of random $p$-tensors of overall
correlation. Our analysis is based on the Malliavin-Stein approach.
