Singular value distribution of dense random matrices with block Markovian dependence

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.

arXiv comment55 pages, 10 figures

References in corpus (26)

• Statistical inference on random dot product graphs: a survey
• Random unistochastic matrices
• A note on a Marčenko-Pastur type theorem for time series
• Spectral properties of complex networks
• Spectrum of large random reversible Markov chains: Heavy-tailed weights on the complete graph
• On the Marčenko-Pastur law for linear time series
• Circular Law Theorem for Random Markov Matrices
• Large sample behaviour of high dimensional autocovariance matrices
• Non-Hermitian random matrices with a variance profile (I): Deterministic equivalents and limiting ESDs
• The Dirichlet Markov Ensemble
• Eigenvalues of block structured asymmetric random matrices
• Limiting spectral distribution of a new random matrix model with dependence across rows and columns
• A Graphon Approach to Limiting Spectral Distributions of Wigner-type Matrices
• Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application
• The ensemble of random Markov matrices
• From random matrices to long range dependence
• Spectrum of large random reversible Markov chains: two examples
• Breakdown of random matrix universality in Markov models
• Bernstein type inequality for a class of dependent random matrices
• Spectrum of large random Markov chains: heavy-tailed weights on the oriented complete graph
• The limiting spectral distribution in terms of spectral density
• Time-inhomogeneous random Markov chains
• Random Band and Block Matrices with Correlated Entries
• On eigenvalue distributions of large auto-covariance matrices
• Random stochastic matrices from classical compact Lie groups and symmetric spaces
• Spectral norm bounds for block Markov chain random matrices

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