  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.
