General mixing time bounds for finite Markov chains via the absolute spectral gap

Abstract

We prove an upper bound on the total variation mixing time of a finite Markov chain in terms of the absolute spectral gap and the number of elements in the state space. Unlike results requiring reversibility or irreducibility, this bound is finite whenever the chain converges. The dependence on the number of elements means that the new bound cannot capture the behavior of rapidly mixing chains; but an example shows that such dependence is necessary. We also provide a sharpened result for reversible chains that are not necessarily irreducible. The proof of the general bound exploits a connection between linear recurrence relations and Schur functions due to Hou and Mu, while the sharpened bound for reversible chains arises as a consequence of the stochastic interpretation of eigenvalues developed by Diaconis, Fill, and Miclo. In particular, for every reversible chain with nonnegative eigenvalues $\beta_j$ we find a strong stationary time whose law is the sum of independent geometric random variables having mean $1/(1-\beta_j)$.