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 \({L_\infty^V}\) , 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 \({L_\infty^V}\) . If the chain is reversible, the same equivalence holds with L 2 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 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 equilibrium is also briefly discussed.
References
Bradley R.C. Jr.: Information regularity and the central limit question. Rocky Mountain J. Math. 13(1), 77–97 (1983)
Brémaud P.: Markov chains: Gibbs fields, Monte Carlo simulation, and queues, volume 31 of Texts in Applied Mathematics. Springer-Verlag, New York (1999)
Brooks S.P., Roberts G.O.: Convergence assessment techniques for Markov chain Monte Carlo. Statistics and Computing 8, 319–335 (1998)
Chen Mu-Fa.: Eigenvalues, inequalities, and ergodic theory, Probability and its Applications (New York). Springer-Verlag London Ltd., London (2005)
Conway J.B.: A course in functional analysis 2nd edn. Springer-Verlag, New York (1990)
Dellnitz M., Junge O.: On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36(2), 491–515 (1999)
Diaconis, P.: Mathematical developments from the analysis of riffle shuffling. In: Groups, combinatorics & geometry (Durham, 2001), pp. 73–97. World Sci. Publ., River Edge NJ, (2003)
Diaconis P.: The Markov chain Monte Carlo revolution. Bull. Amer. Math. Soc. (N.S.) 46(2), 179–205 (2009)
Diaconis P., Holmes S., Neal R.M.: Analysis of a nonreversible Markov chain sampler. Ann. Appl. Probab. 10(3), 726–752 (2000)
Diaconis, P., Khare, K., Saloff-Coste, L.: Gibbs sampling, exponential families and orthogonal polynomials. Statist. Sci., 23(2), pp. 151–200 (2008). (With comments and a rejoinder by the authors)
Diaconis P., Saloff-Coste L.: Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3(3), 696–730 (1993)
Diaconis, P., Saloff-Coste, L.: What do we know about the Metropolis algorithm? J. Comput. System Sci., 57(1), pp. 20–36 (1998). 27th Annual ACM Symposium on the Theory of Computing (STOC’95) (Las Vegas, NV)
Diaconis P., Stroock D.: Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1(1), 36–61 (1991)
Dobrow, R.P., Fill, J.A.: The move-to-front rule for self-organizing lists with Markov dependent requests. In: Discrete probability and algorithms (Minneapolis, MN, 1993), volume 72 of IMA Vol. Math. Appl., pp. 57–80. Springer, New York, (1995)
Evans W., Kenyon C., Peres Y., Schulman L.J.: Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10(2), 410–433 (2000)
Fill J.A.: Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. Ann. Appl. Probab. 1(1), 62–87 (1991)
Häggström O.: On the central limit theorem for geometrically ergodic Markov chains. Probab. Theory Relat. Fields 132(1), 74–82 (2005)
Häggström, O.: Acknowledgement of priority concerning “On the central limit theorem for geometrically ergodic Markov chains” [Probab. Theory Relat. Fields 132(1), 74–82 (2005)]. Probab. Theory Relat. Fields 135(3):470 (2006)
Huisinga, W., Meyn, S.P., Schuette, C.: Phase transitions and metastability in Markovian and molecular systems. Ann. Appl. Probab. (2001). (in press)
Kontoyiannis I., Meyn S.P.: Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13, 304–362 (2003)
Kontoyiannis I., Meyn S.P.: Large deviation asymptotics and the spectral theory of multiplicatively regular Markov processes. Electron. J. Probab. 10(3), 61–123 (2005)
Lawler G.F., Sokal A.D.: Bounds on the L 2 spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Trans. Amer. Math. Soc. 309(2), 557–580 (1988)
Lebeau, G., Diaconis, P.: Métropolis: le jour où l’étoile probabilité entra dans le champ gravitationnel de la galaxie microlocale. In: Séminaire: Équations aux D érivées Partielles. 2006–2007, Sémin. Équ. Dériv. Partielles, pages Exp. No. XIV, 13. École Polytech., Palaiseau (2007)
Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability 2nd edn. Cambridge University Press, London (2009). Published in the Cambridge Mathematical Library. 1993 edition online: http://black.csl.uiuc.edu/~meyn/pages/book.html
Meyn S.P., Tweedie R.L.: Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab. 4(4), 981–1011 (1994)
Montenegro R., Tetali P.: Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci. 1(3), 1–121 (2006)
Nummelin E.: General Irreducible Markov Chains and Nonnegative Operators. Cambridge University Press, Cambridge (1984)
Phatarfod R.M., Pryde A.J., Dyte D.: On the move-to-front scheme with Markov dependent requests. J. Appl. Probab. 34(3), 790–794 (1997)
Roberts G.O., Rosenthal J.S.: Geometric ergodicity and hybrid Markov chains. Electron. Comm. Probab. 2(2), 13–25 (1997) (electronic)
Roberts, G.O., Tweedie, R.L.: Geometric L 2 and L 1 convergence are equivalent for reversible Markov chains. J. Appl. Probab. 38A, 37–41 (2001). Probability, statistics and seismology
Rosenthal J.S.: Correction: Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90(431), 1136 (1995)
Rosenthal J.S.: Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90(430), 558–566 (1995)
Winkler G.: Image Analysis, Random Fields and Dynamic Monte Carlo Methods: A Mathematical Introduction. Springer-Verlag, Berlin (1995)
Wübker, A.: L 2-spectral gaps for time discrete reversible Markov chains. Preprint (2009)
Wübker, A., Kabluchko, Z.: L 2-spectral gaps, weak-reversible and very weak-reversible Markov chains. Preprint (2009)
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I. Kontoyiannis was supported in part by a Sloan Foundation Research Fellowship, and by a Marie Curie International Outgoing Fellowship, PIOF-GA-2009-235837. S.P. Meyn was supported in part by the National Science Foundation ECS-0523620, and by AFOSR grant FA9550-09-1-0190. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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Kontoyiannis, I., Meyn, S.P. Geometric ergodicity and the spectral gap of non-reversible Markov chains. Probab. Theory Relat. Fields 154, 327–339 (2012). https://doi.org/10.1007/s00440-011-0373-4
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DOI: https://doi.org/10.1007/s00440-011-0373-4
Keywords
- Markov chain
- Geometric ergodicity
- Spectral theory
- Stochastic Lyapunov function
- Reversibility
- Spectral gap
Mathematics Subject Classification (2000)
- 60J05
- 60J10
- 37A30
- 37A25