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.

翻译：块马尔可夫链是一种马尔可夫链，其状态空间可划分为有限个聚类，使得转移概率仅依赖于这些聚类。因此，块马尔可夫链作为具有社区结构的马尔可夫链模型。本文建立了当样本路径长度为$\Theta(n^2)$（其中$n$为状态空间大小）时，与块马尔可夫链样本路径相关的经验转移矩阵和经验频率矩阵的奇异值分布的极限定律。证明方法分为两部分。首先，我们引入一类具有依赖项的对称随机矩阵，称为具有方差轮廓的近似不相关随机矩阵，并通过矩方法建立其极限特征值分布。其次，我们发展了一种耦合论证，以证明该通用结果适用于块马尔可夫链相关的奇异值分布。