A CLT in Stein's distance for generalized Wishart matrices and higher order tensors

TL;DR

This paper establishes a high-dimensional central limit theorem for sums of tensor powers of independent vectors, using optimal transport and Stein's method, with implications for various measure classes.

Contribution

It introduces a novel threshold for convergence of tensor power sums to Gaussian distributions in high dimensions, extending existing results.

Findings

Convergence occurs when n^{2p-1} .

Applicable to symmetric log-concave and product measures.

Uses a new optimal transport approach in Stein's method.

Abstract

We study the 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 $n^{2p-1}\gg d$, 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 $X_i^{\otimes p}$.