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

By Ioannis Kontoyiannis, Sean P. Meyn. 2009

113 citations recorded by the source corpus.

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 $${L_\infty^V}$$ , 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 $${L_\infty^V}$$ . If the chain is reversible, the same equivalence holds with L2 in place of $${L_\infty^V}$$ . 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 L2. Moreover, if a chain admits a spectral gap in L2, then for any $${h\in L_2}$$ there exists a Lyapunov function $${V_h\in L_1}$$ such that Vh 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 L2, and the rate at which the chain converges to equilibrium is also briefly discussed.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/01b1c7faa5c4b5ac6b983bab7757ee6a0ecd0e73

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Citations in the corpus

Concentration inequalities for Markov chains by Marton couplings and spectral methods (2012, background): “h reversible, and non-reversible chains, we deﬁne the absolute spectral gap as γ∗ := 1−sup{|λ| : λ∈ S 2,λ6= 1 } if eigenvalue 1 has multiplicity 1, (1.13) γ∗ := 0 otherwise. (1.14) Proposition 1.2 of Kontoyiannis and Meyn (2012) shows that for φ-irreducible, aperiodic, reversible chains, geometric e”
A non‐conservative Harris ergodic theorem (2019, background): “Under aperiodicity condition, these assumptions ensure that Mτ admits a unique invariant probability measure γ and satisfies a contraction principle for the weighted total variation distance (see [65, 74, 75]).”