A central limit theorem for tensor powers

We introduce the Wishart tensor as the p-th tensor power of a given random vector X in R^n. This is inspired by the classical Wishart matrix, obtained when p=2. Sums of independent Wishart tensors appear naturally when studying random geometric graphs as well as universality phenomena in large neural networks and we will discuss possible connections and recent results.

The main focus of the talk will be quantitative estimates for the central limit theorem of Wishart tensors. In this setting, we will explain how Stein's method may be used to exploit the low dimensional structure which is inherent to tensor powers. Specifically, it will be shown that, under appropriate regularity assumptions, a sum of independent Wishart tensors is close to a Gaussian tensor as soon as n^(2p-1)<<d. Here, n is the dimension of the random vector and d is the number of independent copies.