Marchenko-Pastur law for a random tensor model

TL;DR

This paper investigates the spectral distribution of large covariance matrices derived from symmetric random tensors, establishing conditions under which it follows the Marchenko-Pastur law as the tensor dimension grows.

Contribution

It provides the first optimal conditions for the Marchenko-Pastur law to hold for covariance matrices from symmetric random tensors with increasing order.

Findings

Marchenko-Pastur law holds when $d^2=o(n)$ for bounded fourth moments.

Introduces a new concentration inequality for quadratic forms in symmetric tensors.

Establishes a law of large numbers for elementary symmetric random polynomials.

Abstract

We study the limiting spectral distribution of large-dimensional sample covariance matrices associated with symmetric random tensors formed by $\binom{n}{d}$ different products of $d$ variables chosen from $n$ independent standardized random variables. We find optimal sufficient conditions for this distribution to be the Marchenko-Pastur law in the case $d=d(n)$ and $n\to\infty$. Our conditions reduce to $d^2=o(n)$ when the variables have uniformly bounded fourth moments. The proofs are based on a new concentration inequality for quadratic forms in symmetric random tensors and a law of large numbers for elementary symmetric random polynomials.