  We study the fluctuations, as $d,n\to \infty$, of the Wishart matrix
$\mathcal{W}_{n,d}= \frac{1}{d} \mathcal{X}_{n,d} \mathcal{X}_{n,d}^{T} $
associated to a $n\times d$ random matrix $\mathcal{X}_{n,d}$ with non-Gaussian
entries. We analyze the limiting behavior in distribution of
$\mathcal{W}_{n,d}$ in two situations: when the entries of $\mathcal{X}_{n,d}$
are independent elements of a Wiener chaos of arbitrary order and when the
entries are partially correlated and belong to the second Wiener chaos. In the
first case, we show that the (suitably normalized) Wishart matrix converges in
distribution to a Gaussian matrix while in the correlated case, we obtain its
convergence in law to a diagonal non-Gaussian matrix. In both cases, we derive
the rate of convergence in the Wasserstein distance via Malliavin calculus and
analysis on Wiener space.
