Mixing Time Estimation in Ergodic Markov Chains from a Single Trajectory with Contraction Methods

The mixing time $t_{\mathsf{mix}}$ of an ergodic Markov chain measures the rate of convergence towards its stationary distribution $\boldsymbol{\pi}$. We consider the problem of estimating $t_{\mathsf{mix}}$ from one single trajectory of $m$ observations $(X_1, …, X_m)$, in the case where the transition kernel $\boldsymbol{M}$ is unknown, a research program started by Hsu et al. [2015]. The community has so far focused primarily on leveraging spectral methods to estimate the