# Geometric Ergodicity and the Spectral Gap of Non-Reversible Markov Chains - Mathematics > Probability

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Abstract: We argue that the spectral theory of non-reversible Markov chains may oftenbe more effectively cast within the framework of the naturally associatedweighted-$L \infty$ space $L \infty^V$, instead of the usual Hilbert space$L 2=L 2\pi$, where $\pi$ is the invariant measure of the chain. Thisobservation is, in part, based on the following results. A discrete-time Markovchain with values in a general state space is geometrically ergodic if and onlyif its transition kernel admits a spectral gap in $L \infty^V$. If the chain isreversible, the same equivalence holds with $L 2$ in place of $L \infty^V$, butin the absence of reversibility it fails: There are necessarilynon-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 equilibriumis also briefly discussed.

Author: ** Ioannis Kontoyiannis, Sean P. Meyn**

Source: https://arxiv.org/