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Spectral norm bounds for block Markov chain random matrices

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  • Sanders, Jaron
  • Senen–Cerda, Albert

Abstract

This paper quantifies the asymptotic order of the largest singular value of a centered random matrix built from the path of a Block Markov Chain (BMC). In a BMC there are n labeled states, each state is associated to one of K clusters, and the probability of a jump depends only on the clusters of the origin and destination. Given a path X0,X1,…,XTn started from equilibrium, we construct a random matrix Nˆ that records the number of transitions between each pair of states. We prove that if ω(n)=Tn=o(n2), then ‖Nˆ−E[Nˆ]‖=ΩP(Tn/n). We also prove that if Tn=Ω(nlnn), then ‖Nˆ−E[Nˆ]‖=OP(Tn/n) as n→∞; and if Tn=ω(n), a sparser regime, then ‖NˆΓ−E[Nˆ]‖=OP(Tn/n). Here, NˆΓ is a regularization that zeroes out entries corresponding to jumps to and from most-often visited states. Together this establishes that the order is ΘP(Tn/n) for BMCs.

Suggested Citation

  • Sanders, Jaron & Senen–Cerda, Albert, 2023. "Spectral norm bounds for block Markov chain random matrices," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 134-169.
    • Handle: RePEc:eee:spapps:v:158:y:2023:i:c:p:134-169
    DOI: 10.1016/j.spa.2022.12.004
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    References listed on IDEAS

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    1. Winfried Hochstättler & Werner Kirsch & Simone Warzel, 2016. "Semicircle Law for a Matrix Ensemble with Dependent Entries," Journal of Theoretical Probability, Springer, vol. 29(3), pages 1047-1068, September.
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