A CLT in Stein’s Distance for Generalized Wishart Matrices and Higher-Order Tensors

Abstract We study the a central limit theorem for sums of independent tensor powers, $\frac{1}{\sqrt{d}}\sum \limits _{i=1}^d X_i^{\otimes p}$. We focus on the high-dimensional regime where $X_i \in{\mathbb{R}}^n$ and $n$ may scale with $d$. Our main result is a proposed threshold for convergence. Specifically, we show that, under some regularity assumption, if $n^{2p-1}\ll d$, then the normalized sum converges to a Gaussian. The results apply, among others, to symmetric uniform log-concave measures and to product measures. This generalizes several results found in the literature. Our main technique is a novel application of optimal transport to Stein’s method, which accounts for the low-dimensional structure, which is inherent in $X_i^{\otimes p}$.