Marchenko-Pastur law for a random tensor model

Abstract

We study the limiting spectral distribution of large-dimensional sample covariance matrices associated with symmetric random tensors formed by 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 → ∞. 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.