Sharp concentration for sums of matrices with Markovian dependence through universality

TL;DR

This paper establishes a universality principle for sums of matrices generated by Markov chains, showing they share spectral properties with Gaussian matrices and enabling sharp concentration inequalities.

Contribution

It introduces a nonasymptotic universality result for Markovian matrix sums, extending Gaussian concentration techniques to dependent structures.

Findings

Spectral properties of Markovian matrix sums match Gaussian matrices.

Enables polynomial dimensional improvements in concentration bounds.

Recovers sharp limiting values for spectral clustering models.

Abstract

We prove that a sum of random matrices generated by a $\psi$-mixing Markov chain has similar spectral properties to a Gaussian matrix with the same mean and covariance structure. This nonasymptotic universality principle enables sharp concentration inequalities when combined with recent advances in the Gaussian literature. We illustrate the theory with examples, showing how it enables polynomial dimensional improvements relative to previous Markovian matrix concentration results when applied to Wigner-type matrices, and how one can recover sharp limiting values for a model used to study spectral clustering techniques. A key challenge in the proof is that techniques based only on classical cumulants, which can be used when summands are independent, are not sufficient on their own for efficient estimates in a Markovian setting. Our approach exploits Boolean cumulants and a change--of--measure argument.