Singular value distribution of dense random matrices with block Markovian dependence

TL;DR

This paper analyzes the singular value distribution of matrices derived from block Markov chains, establishing limiting laws for large sample paths and introducing a new class of dependent random matrices.

Contribution

It introduces approximately uncorrelated random matrices with variance profiles and applies them to derive singular value distribution laws for block Markov chains.

Findings

Established limiting eigenvalue distributions for a new class of random matrices.

Proved convergence of singular value distributions for block Markov chain matrices.

Developed a coupling argument to connect general random matrices to block Markov chain matrices.

Abstract

A block Markov chain is a Markov chain whose state space can be partitioned into a finite number of clusters such that the transition probabilities only depend on the clusters. Block Markov chains thus serve as a model for Markov chains with communities. This paper establishes limiting laws for the singular value distributions of the empirical transition matrix and empirical frequency matrix associated to a sample path of the block Markov chain whenever the length of the sample path is $\Theta(n^2)$ with $n$ the size of the state space. The proof approach is split into two parts. First, we introduce a class of symmetric random matrices with dependent entries called approximately uncorrelated random matrices with variance profile. We establish their limiting eigenvalue distributions by means of the moment method. Second, we develop a coupling argument to show that this general-purpose result applies to the singular value distributions associated with the block Markov chain.