On spectrum of sample correlated matrices from large fold tensor vectors

TL;DR

This paper studies the spectral distribution of sample correlation matrices formed from large tensor product vectors, showing it converges to the Marčenko-Pastur law under certain asymptotic conditions.

Contribution

It establishes the limiting spectral distribution for sample correlation matrices from tensor product vectors, extending the Marčenko-Pastur law to this setting.

Findings

Spectral distribution converges to Marčenko-Pastur law as n,k → ∞ with k = o(n)

Limiting distribution applies to Wishart matrices from tensor products of uniform vectors

Results hold for large-dimensional tensor product sample vectors

Abstract

In this paper, we investigate the limiting spectral distribution of the sample correlation matrix, whose sample vectors are $k$-fold tensor products of $n$-dimensional vectors with i.i.d. entries. We focus on the limiting regime $n,k \to \infty$ with $k = o(n)$, and we show that the limiting spectral distribution is the Mar\v{c}enko-Pastur law. As a consequence, we show that the limiting spectral distribution of the Whishart matrix from the $k$-fold tensor product of independent uniformly distributed unit vectors in $\mathbb C^n$ is the Mar\v{c}enko-Pastur law.