Geometric Ergodicity and the Spectral Gap of Non-Reversible Markov Chains

TL;DR

This paper explores the spectral properties of non-reversible Markov chains, showing that their geometric ergodicity is better characterized in a weighted $L_ty^V$ space rather than the traditional $L_2$ space, revealing new insights into their convergence behavior.

Contribution

It establishes the equivalence between geometric ergodicity and the existence of a spectral gap in $L_ty^V$, and highlights differences from the reversible case, providing a new framework for analyzing non-reversible chains.

Findings

Spectral gap in $L_ty^V$ characterizes geometric ergodicity.

Non-reversible chains can have a spectral gap in $L_ty^V$ but not in $L_2$.

Existence of a Lyapunov function $V_h$ linking $L_2$ spectral gap to $L_ty^{V_h}$ spectral gap.

Abstract

We argue that the spectral theory of non-reversible Markov chains may often be more effectively cast within the framework of the naturally associated weighted-$L_\infty$ space $L_\infty^V$, instead of the usual Hilbert space $L_2=L_2(\pi)$, where $\pi$ is the invariant measure of the chain. This observation is, in part, based on the following results. A discrete-time Markov chain with values in a general state space is geometrically ergodic if and only if its transition kernel admits a spectral gap in $L_\infty^V$. If the chain is reversible, the same equivalence holds with $L_2$ in place of $L_\infty^V$, but in the absence of reversibility it fails: There are (necessarily non-reversible, geometrically ergodic) chains that admit a spectral gap in $L_\infty^V$ but not in $L_2$. Moreover, if a chain admits a spectral gap in $L_2$, then for any $h\in L_2$ there exists a Lyapunov function $V_h\in L_1$ such that $V_h$ dominates $h$ and the chain admits a spectral gap in $L_\infty^{V_h}$. The relationship between the size of the spectral gap in $L_\infty^V$ or $L_2$, and the rate at which the chain converges to equilibrium is also briefly discussed.