Advertisement

Journal of Theoretical Probability

, Volume 9, Issue 2, pp 459–510 | Cite as

Nash inequalities for finite Markov chains

  • P. Diaconis
  • L. Saloff-Coste
Article

Abstract

This paper develops bounds on the rate of decay of powers of Markov kernels on finite state spaces. These are combined with eigenvalue estimates to give good bounds on the rate of convergence to stationarity for finite Markov chains whose underlying graph has moderate volume growth. Roughly, for such chains, order (diameter) steps are necessary and suffice to reach stationarity. We consider local Poincaré inequalities and use them to prove Nash inequalities. These are bounds onl2-norms in terms of Dirichlet forms andl1-norms which yield decay rates for iterates of the kernel. This method is adapted from arguments developed by a number of authors in the context of partial differential equations and, later, in the study of random walks on infinite graphs. The main results do not require reversibility.

Key Words

Markov chains Dirichlet forms infinite graphs Nash inequalities 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bakry, D., Coulhon, T., Ledoux, M., and Saloff-Coste, L. (1995). Sobolev inequalities in disguise. Université Paul Sabtier, Toulouse.Stat. et. Prob., Preprint.Google Scholar
  2. 2.
    Carlen, E., Kusuoka, S., and Stroock, D. (1987). Upper bounds for symmetric Markov transition functions.Ann. Inst. H. Poincaré, Prob. Stat. 23, 245–287.Google Scholar
  3. 3.
    Chung, F., Diaconis, P., and Graham, R. (1987). A random walk problem arising in random number generation.Ann. Prob. 15, 1148–1165.Google Scholar
  4. 4.
    Coulhon, T., and Saloff-Coste, L. (1990a). Puissance d'un opérateur regularisant.Ann. Inst. H. Poincaré, Prob. Stat. 26, 419–436.Google Scholar
  5. 5.
    Coulhon, T., and Saloff-Coste, L. (1990b). Marches aléatoires non-symétriques sur les groupes unimodulaires.C. R. Acad. Sci. Paris, Série I,310, 627–630.Google Scholar
  6. 6.
    Coulhon, T., and Saloff-Coste, L. (1993). Isopérimétrie sur les groupes et les variétés.Rev. Mat. Iberoamericana,9, 293–314.Google Scholar
  7. 7.
    Diaconis, P. (1982).Applications of Non-Commutative Fourier Analysis to Probability Problems. Springer L.N.M.1362, 51–100.Google Scholar
  8. 8.
    Diaconis, P. (1986).Group Representations in Probability and Statistics. IMS, Hayward.Google Scholar
  9. 9.
    Diaconis, P., and Saloff-Coste, L. (1933a). Comparison theorems for reversible Markov chains.Ann. Appl. Prob. 3, 696–730.Google Scholar
  10. 10.
    Diaconis, P., and Saloff-Coste, L. (1993b). Comparison techniques for random walk on finite groups.Ann. Prob. 21, 2131–2156.Google Scholar
  11. 11.
    Diaconis, P., and Saloff-Coste, L. (1994a). Moderate growth and random walk on finite groups.Geom. and Funct. Anal. 4, 1–34.Google Scholar
  12. 12.
    Diaconis, P., and Saloff-Coste, L. (1995). An application of Harnack inequalities to random walk on nilpotent quotients. Actes du Colloque en l'honneur de J. P. Kahane.J. Fourier Anal. Appl. (special Kahane issue) 189–208.Google Scholar
  13. 13.
    Diaconis, P., and Saloff-Coste, L. (1992). Logarithmic Sobolev inequalities and finite Markov chains.Annals Apol. Prob. (to appear).Google Scholar
  14. 14.
    Diaconis, P., and Saloff-Coste, L. (1994c). Geometric bounds on character ratios and symmetric functions. Manuscript.Google Scholar
  15. 15.
    Diaconis, P., and Shashahani, M. (1981). Generating a random permutation with random transpositions.Z. Wahrsch. Verw. Geb. 57, 159–179.Google Scholar
  16. 16.
    Diaconis, P., and Stroock, D. (1991). Geometric bounds for eigenvalues for Markov chains.Ann. Appl. Prob. 1, 36–61.Google Scholar
  17. 17.
    Diaconis, P., and Sturmfels, B. (1993). Algebraic algorithms for sampling from conditional distributions.Ann. Stat. (to appear).Google Scholar
  18. 18.
    Dyer, M., and Frieze, A. (1991). Computing the volume of convex bodies: a case where randomness proveably helps. In Bollobàs, B. (ed.),Probabilistic Combinatorics and Its Applications, Proc. Symp. Appl. Math. 44, 123–170.Google Scholar
  19. 19.
    Fabes, E. (1993). Gaussian upper bounds on fundamental solutions of parabolic equations: the method of Nash. InDirichlet Forms, L.N.M. Springer, p. 163.Google Scholar
  20. 20.
    Feller, W. (1968).An Introduction to Probability Theory and Its Applications, Vol. I, Third Edition J. Wiley and Sons.Google Scholar
  21. 21.
    Fill, J. (1991). Eigenvalue bounds on convergence to stationarity for nonreveresible Markov chains with an application to the exclusion process.Ann. Appl. Prob. 1, 62–87.Google Scholar
  22. 22.
    Gohberg, I., and Krein, I. (1969). Introduction to the theory of linear non-selfadjoint operators. Providence.Amer. Math. Soc. Google Scholar
  23. 23.
    Hildebrand, M. (1992). Generating random elements in SLn(F q) by random transvections.J. Alg. Combinatorics 1, 133–150.Google Scholar
  24. 24.
    Horn, R., and Johnson, C. (1990).Topics in Matrix Analysis. Cambridge University Press.Google Scholar
  25. 25.
    Jerrum, M., and Sinclair, A. (1989). Approximating the permanent.SIAM J. Comp. 18, 1149–1178.Google Scholar
  26. 26.
    Lawler, G., and Sokal, A. (1988). Bounds on theL 2 spectrum for Markov chains and Markov processes: A generalization of Cheeger's inequality.Trans. Amer. Math. Soc. 309, 557–580.Google Scholar
  27. 27.
    Lovasz, L., and Simonovits, M. (1990). Random walks in a convex body and an improved volume algorithm.Random Struct. and Algo. 4, 359–412.Google Scholar
  28. 28.
    Mihail, E. (1989). Combinatorial aspects of expanders. Ph.D. dissertation, Department Computer Sciences, Harvard University.Google Scholar
  29. 29.
    Nash, J. (1958). Continuity of solutions of parabolic and elliptic equations.Amer. J. Math. 80, 931–954.Google Scholar
  30. 30.
    Marshall, A., and Olkin, I. (1979).Inequalities: The Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
  31. 31.
    Robinson, D. (1991).Elliptic Operators on Lie Groups. Oxford University Press.Google Scholar
  32. 32.
    Sinclair, A., and Jerrum, M. (1989). Approximate counting, uniform generation and rapidly mixing Markov chains.Inform. and Comput. 82, 93–133.Google Scholar
  33. 33.
    Sinclair, A. (1993).Algorithms for Random Generation and Counting: A Markov Chain Approach. Birkhäuser, Boston.Google Scholar
  34. 34.
    Stein, E., and Weiss, G. (1971).Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press.Google Scholar
  35. 35.
    Varopoulos, N. (1984). Une généralisation du théorème de Hardy-Littlewood-Sobolev pour les espaces de Dirichlet.C. R. Acad. Sc. Paris, Série I,299, 651–654.Google Scholar
  36. 36.
    Varopoulos, N. (1985). Isoperimetric inequalities and Markov chains.J. Funct. Anal. 63, 215–239.Google Scholar
  37. 37.
    Varopoulos, N., Saloff-Coste, L., and Coulhon, T. (1993).Analysis and Geometry on Groups. Cambridge University Press.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • P. Diaconis
    • 1
  • L. Saloff-Coste
    • 2
  1. 1.Department of MathematicsHarvard UniversityCambridge
  2. 2.CNRSUniversité Paul Sabatier, Laboratoire Statistique et ProbabilitésToulouse CedexFrance

Personalised recommendations