## 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 Θ(n2) 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.

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.

Original language | English |
---|---|

Pages (from-to) | 453-504 |

Number of pages | 52 |

Journal | Stochastic Processes and their Applications |

Volume | 158 |

DOIs | |

Publication status | Published - 1 Apr 2023 |

### Funding

This work is part of the project Clustering and Spectral Concentration in Markov Chains with project number OCENW.KLEIN.324 of the research programme Open Competition Domain Science – M which is partly financed by the Dutch Research Council (NWO) . We would like to thank Sem Borst, Martijn Gösgens, Gianluca Kosmella, Albert Senen–Cerda and Haodong Zhu for providing feedback on a draft of this manuscript.

Funders | Funder number |
---|---|

Nederlandse Organisatie voor Wetenschappelijk Onderzoek |

## Keywords

- Approximately uncorrelated
- Block Markov chains
- Poisson limit theorem
- Random matrices
- Variance profile