Limit Behavior in High-Dimensional Regime for the Wishart Tensors in Wiener Chaos

We analyze the p-random Wishart tensor ( \(p\ge 2\) $p\ge 2$ ) associated with an initial \(n\times d\) $n\times d$ random matrix \(\mathcal {X}_{n,d}\) ${\mathcal{X}}_{n,d}$ whose entries are independent and belong to the Wiener chaos. The order of the Wiener chaos is constant in each row of the matrix, and it may change from one row to another. Therefore, the entries of our starting matrix are not identically distributed (we only assume that their second moments coincide). We prove that, in high-dimensional regime (i.e. when n and d are large enough), the multidimensional random vector corresponding to the p-Wishart tensor is close in distribution to a standard multidimensional Gaussian vector. We also evaluate the Wasserstein distance between the probability distributions of these two random vectors. By using the techniques of the Stein–Malliavin calculus, we show that when the dimensions n and d are large enough and p is fixed, this Wasserstein distance is of order less than \(\sqrt{\frac{n^{2p-1}}{d}}\) $\sqrt{\frac{n^{2p-1}}{d}}$ .