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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-Gaussianentries. 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 theentries are partially correlated and belong to the second Wiener chaos. In thefirst case, we show that the (suitably normalized) Wishart matrix converges indistribution to a Gaussian matrix while in the correlated case, we obtain itsconvergence in law to a diagonal non-Gaussian matrix. In both cases, we derivethe rate of convergence in the Wasserstein distance via Malliavin calculus andanalysis on Wiener space.

Limiting behavior of large correlated Wishart matrices with chaotic entries