2009/06/29 by Ioannis Kontoyiannis, Sean Meyn, Kontoyiannis, Ioannis +1 · 2 citations
Mathematics · #37A25 #37A30 #60J05 #60J10 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #Probability (math.PR) #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.0906.5322
openalex publication_date 2009/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, but in the absence of reversibility it fails: There are (necessarily non-reversible, geometrically ergodic) chains that admit a spectral gap in L_∞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.