Asymptotic freeness of sample covariance matrices via embedding

TL;DR

This paper provides a new proof demonstrating that independent sample covariance matrices become asymptotically free as their dimension and sample size grow proportionally, by embedding them into larger Wigner matrices.

Contribution

It introduces an alternative proof method for asymptotic freeness of sample covariance matrices using embedding into Wigner matrices and their known properties.

Findings

Sample covariance matrices are asymptotically free when dimension and sample size grow proportionally.

Embedding into Wigner matrices simplifies the proof of asymptotic freeness.

The approach leverages known results about Wigner matrices and deterministic matrices.

Abstract

We present an alternative proof of asymptotic freeness of independent sample covariance matrices, when the dimension and the sample size grow at the same rate, by embedding these matrices into Wigner matrices of a larger order and using asymptotic freeness of independent Wigner and deterministic matrices.