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 dependence 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 block Markov chains.

翻译：Markov 链条是一个区块 Markov 链条, 其状态空间可以分割成数量有限的组群, 使过渡概率仅取决于组群。 区块 Markov 链条因此成为Markov 链条和社区的模型。 本文为在区块 Markov 链条的单值分布设置了限制法律。 当区块 Markov 链条的样本路径长度为$\ Theta (n)2美元, 以国家空间的大小为美元时, 与该区块的样本路径相关联的实验频率矩阵。 校对方法分为两部分。 首先, 我们引入了一组依赖性的对称随机矩阵, 称为大致不相干随机矩阵, 并带有差异剖面。 我们通过瞬间方法来设定其限制的单值分布。 其次, 我们开发一个组合参数, 以显示这一通用结果适用于 区块链 。