Three Kinds of Geometric Convergence for Markov Chains and the Spectral Gap Property

Wolfgang Stadje (University of Osnabrück)
Achim Wübker (University of Osnabrück)

Abstract


In this paper we investigate three types of convergence for geometrically ergodic Markov chains (MCs) with countable state space, which in general lead to different `rates of convergence'. For reversible Markov chains it is shown that these rates coincide. For general MCs we show some connections between their rates and those of the associated reversed MCs. Moreover, we study the relations between these rates and a certain family of isoperimetric constants. This sheds new light on the connection of geometric ergodicity and the so-called spectral gap property, in particular for non-reversible MCs, and makes it possible to derive sharp upper and lower bounds for the spectral radius of certain non-reversible chains

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1001-1019

Publication Date: April 20, 2011

DOI: 10.1214/EJP.v16-900

References

  1. Cogburn, R. The central limit theorem for Markov processes. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 (1972), 485--512. Math. Review number not available
  2. Döblin, W. Sur les propriÈtÈs asymptotiques de mouvement rÈgis par certains types de chaÓnes simples. Bull. Math. Soc. Roum. Sci. 39 (1937), 57--115. Math. Review number not available
  3. Häggström, O. On the central limit theorem for geometrically ergodic Markov chains. Probab. Th. Relat. Fields 132 (2005), 74--82. Math. Review 2005m:60155
  4. Kontoyiannis, I., Meyn, S.P. Geometric ergodicity and the spectral gap of non-reversible Markov chains. arXiv 0906.5322 (2009). Math. Review number not available.
  5. Lawler, G.F., Sokal, A.D. Bounds on the L2 spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Trans. Amer. Math. Soc. 309 (1988), 557--580. Math. Review 89h:60105
  6. Levin, D.A., Peres, Y., Wilmer, E.L. Markov Chains and Mixing Times (2008), American Mathematical Society. Math. Review 2010c:60209
  7. Meyn, S.P., Tweedie, R.L. Markov Chains and Stochastic Stability (1993), Springer. Math Review 95j:60103
  8. Nummelin, E. General Irreducible Markov Chains and Non-Negative Operators (1984), Cambridge Univ. Press. Math. Review 87a:60074
  9. Nummelin, E., Tuominen, P. Geometric ergodicity of Harris recurrent chains with applications to renewal theory Stoch. Proc. Appl. 12 (1982), 187--202. Math. Review 83f:60089
  10. Reed, M., Simon, B. Methods of Modern Mathematical Physics. Volume I: Functional Analysis (1972), Academic Press. Math. Review 85e:46002
  11. Roberts, G.O., Tweedie, R.L. Geometric L2 and L1 convergence are equivalent for reversible Markov chains. J. Appl. Probab. 38(A) (2001), 37--41. Math. Review 2003g:60122
  12. Roberts, G.O., Rosenthal, J.S. Geometric ergodicity and hybrid Markov chains. Electron. Comm. Probab. 2 (1997), 13--25. Math. Review 99b:60122
  13. Saloff-Coste, L. Lectures on Finite Markov Chains. Lecture Notes in Math. 1665 (1996), Springer. Math. Review 99b:60119
  14. Wübker, A. Asymptotic optimality of isoperimetric constants. To be published in Theoretical Journal of Probability
  15. Wübker, A. L2-spectral gaps for time discrete reversible Markov chains Preprint (2011). Math. Review number not available.
  16. Wübker, A. Spectral theory for weakly reversible Markov chains. Preprint, 2011.
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.