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

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 , instead of the usual Hilbert space L 2 = L 2 (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 . If the chain is reversible, the same equivalence holds with L 2 in place of . In the absence of reversibility it fails: There are (necessarily non-reversible, geometrically ergodic) chains that admit a spectral gap in but not in L 2 . Moreover, if a chain admits a spectral gap in L 2 , then for any there exists a Lyapunov function such that V h dominates h and the chain admits a spectral gap in . The relationship between the size of the spectral gap in or L 2 , and the rate at which the chain converges to equilibrium is also briefly discussed.