Spectral norm bounds for block Markov chain random matrices

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 $X_0, X_1, \ldots, X_{T_n}$ started from equilibrium, we construct a random matrix $\hat{N}$ that records the number of transitions between each pair of states. We prove that if $\omega(n) = T_n = o(n^2)$, then $\| \hat{N} - \mathbb{E}[\hat{N}] \| = \Omega_{\mathbb{P}}(\sqrt{T_n/n})$. We also prove that if $T_n = \Omega(n \ln{n})$, then $\| \hat{N} - \mathbb{E}[\hat{N}] \| = O_{\mathbb{P}}(\sqrt{T_n/n})$ as $n \to \infty$; and if $T_n = \omega(n)$, a sparser regime, then $\| \hat{N}_\Gamma - \mathbb{E}[\hat{N}] \| = O_{\mathbb{P}}(\sqrt{T_n/n})$. Here, $\hat{N}_{\Gamma}$ is a regularization that zeroes out entries corresponding to jumps to and from most-often visited states. Together this establishes that the order is $\Theta_{\mathbb{P}}(\sqrt{T_n/n})$ for BMCs.