Geometric ergodicity and the spectral gap of non-reversible Markov chains

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∞ space L∞V, instead of the usual Hilbert space L2 = L2(π), where π 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∞V. If the chain is reversible, the same equivalence holds with L2 in place of L∞V. In the absence of reversibility it fails: There are (necessarily non-reversible, geometrically ergodic) chains that admit a spectral gap in Linfin;V but not in L2. Moreover, if a chain admits a spectral gap in L2, then for any h ∈ L2 there exists a Lyapunov function Vh ∈ L1 such that Vh dominates h and the chain admits a spectral gap in L∞Vh. The relationship between the size of the spectral gap in L∈V or L2, and the rate at which the chain converges to equilibrium is also briefly discussed. © 2011 Springer-Verlag.