Bernstein type inequality for a class of dependent random matrices

TL;DR

This paper extends Bernstein-type inequalities to sums of dependent self-adjoint random matrices with bounded eigenvalues, providing a new tool for analyzing matrix-valued dependent data.

Contribution

It introduces a Bernstein inequality for dependent random matrices, generalizing previous scalar results to the matrix setting with geometric absolute regularity.

Findings

Provides a Bernstein inequality for dependent random matrices

Uses decoupling of Laplace transforms on Cantor-like sets

Extends scalar inequalities to matrix-valued dependent data

Abstract

In this paper we obtain a Bernstein type inequality for the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality can be viewed as an extension to the matrix setting of the Bernstein-type inequality obtained by Merlev\`ede et al. (2009) in the context of real-valued bounded random variables that are geometrically absolutely regular. The proofs rely on decoupling the Laplace transform of a sum on a Cantor-like set of random matrices.