Journal of Theoretical Probability

, Volume 7, Issue 3, pp 483–497 | Cite as

Uniform ergodic convergence and averaging along Markov chain trajectories

  • Yves Derriennic
  • Michael Lin
Article

Abstract

LetP(x, A) be a transition probability on a measurable space (S, Σ) and letX n be the associated Markov chain.Theorem. LetfB(S, Σ). Then for anyxS we haveP x a.s.
$$\mathop {\underline {\lim } }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {f(X_k ) \geqslant } \mathop {\underline {\lim } }\limits_{n \to \infty } \mathop {\inf }\limits_{x \in S} \frac{1}{n}\sum\limits_{k = 1}^n {P^k f(x)} $$
and (implied by it) a corresponding inequality for the lim. If 1/n k=1 n P k f converges uniformly, then for everyx∈S, 1/n k=1 n f(X k ) convergesP x a.s.

Applications are made to ergodic random walks on amenable locally compact groups. We study the asymptotic behavior of 1/n k=1 n μ k *f and of 1/n k=1 n f(X k ) via that ofΨ n *f(x)=m(A n )−1 An f(xt), where {A n } is a Følner sequence, in the following cases: (i)f is left uniformly continuous (ii) μ is spread out (iii)G is Abelian.

Non-Abelian Example: Let μ be adapted and spread-out on a nilpotent σ-compact locally compact groupG, and let {A n } be a Følner sequence. If forfB(G, ∑) m(A n )−1 An f(xt)dm(t) converges uniformly, then 1/n k=1 n f(X k ) converges uniformly, andP x convergesP x a.s. for everyxG.

Key Words

Ergodic theorems strong law of large numbers Markov chains random walks 

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References

  1. 1.
    Azencott, R. (1970). Espaces de Poisson des groupes localement compactes,Springer Lecture Notes Math., Vol. 148.Google Scholar
  2. 2.
    Breiman, L. (1960). The strong law of large numbers for a class of Markov chains,Ann. Math. Stat. 31, 801–803.Google Scholar
  3. 3.
    Besicovitch, A. S., and Bohr, H. (1931). Almost periodicity and general trigonometric series,Actu Math. 57, 203–292.Google Scholar
  4. 4.
    Derriennic, Y. (1976). Lois “Zéro au deux” pour les processus de Markov. Applications aux marches aléatoires,Ann. Inst. Poincaré B12, 111–129.Google Scholar
  5. 5.
    Derriennic, Y. (1986). Entropie, théorèmes limites et marches aléatoires,Springer Lecture Notes Math. 1210, 241–284.Google Scholar
  6. 6.
    Derriennic, Y. (1988). Entropy and boundary for random walks on locally compact groups,Trans. Tenth Prague Conf. on Information Theory etc., Academia, Prague, pp. 269–275.Google Scholar
  7. 7.
    Derriennic, Y. (1985). Sur le théorème de point fixe de Brunel et le théorème de Choquet-Deny,Ann. Sc. Univ. Chermont-Ferrand II,87, 107–111.Google Scholar
  8. 8.
    Derriennic, Y., and Lin, M. (1989). Convergence of iterates of averages of certain operator representations and of convolution powers,J. Funct. Anal. 85, 86–102.Google Scholar
  9. 9.
    Eberlein, W. F. (1949). Abstract ergodic theorems and weak almost periodic functions,Trans. American Math. Soc. 67, 217–240.Google Scholar
  10. 10.
    Furstenberg, H. (1963). Noncommuting random products,Trans. American Math. Soc. 108, 377–428.Google Scholar
  11. 11.
    Guivarch, Y. (1973). Croissance polynomiale et périodes des fonctions harmoniques,Bull. Soc. Math. France 101, 333–379.Google Scholar
  12. 12.
    Gordin, M. I., and Lifisc, B. A. (1978). The central limit theorem for stationary Markov processes,Doklady 239; English translation:Soviet Math. Doklady 19, 392–393.Google Scholar
  13. 13.
    Horowitz, S. (1972). Transition probabilities and contractions ofL ,Z. Wahrsch. 24, 263–274.Google Scholar
  14. 14.
    Högnäs, G., and Mukherjea, A. (1984). On the limit value of the average of the values of a function at random points,Springer Lecture Notes in Math. 1064, 204–218.Google Scholar
  15. 15.
    Jamison, B. (1965). Ergodic decomposition induced by certain Markov operators,Trans. American Math. Soc. 117, 451–468.Google Scholar
  16. 16.
    Jamison, B., and Sine, R. (1974). Sample path convergence of Markov processes,Z. Wahrsch. 28, 173–177.Google Scholar
  17. 17.
    Krengel, U. (1985).Ergodic Theorems, deGuyter, Berlin.Google Scholar
  18. 18.
    Kahane, J. P., Peyrière, J., Zhi-ying, W., and Li-ming, W. (1988). Moyennes uniformes et moyennes suivant une marche aléatoire,Probability The. Rel. Fields 79, 625–628.Google Scholar
  19. 19.
    Kaimanovich, V. A., and Vershik, A. M. (1983). Random walks on discrete groups: boundary and entropy,Ann. Prob. 11, 457–490.Google Scholar
  20. 20.
    Lin, M., and Sine, R. (1977). On the individual ergodic theorem,Z. Wahrsch. 38, 329–331.Google Scholar
  21. 21.
    Meilijson, I. (1973). The averages of the values of a function at random points,Israel J. Math. 15, 193–203.Google Scholar
  22. 22.
    Neveu, J. (1964).Bases Mathématiques du Calcul des Probabilités, Masson, Paris.Google Scholar
  23. 23.
    Namioka, I., and Asplund, E. (1967). A geometric proof of Ryll-Nardzewski's fixed point theorem,Bull. Amer. Math. Soc. 73, 443–445.Google Scholar
  24. 24.
    Pier, J.-P. (1984).Amenable Locally Compact Groups, John Wiley and Sons, New York.Google Scholar
  25. 25.
    Rosenblatt J. (1981). Ergodic and mixing random walks on locally compact groups,Math. Ann. 257, 31–42.Google Scholar
  26. 26.
    Robbins, H. (1953). On the equidistribution of sums of independent random variables,Proc. Am. Math. Soc. 4, 786–799.Google Scholar
  27. 27.
    Ryll-Nardzewski, C. (1962). Generalized random ergodic theorems and weakly almost periodic functions.Bull. Acad. Polonaise Sci. 10, 271–275.Google Scholar
  28. 28.
    Ryll-Nardzewski, C. (1966). On fixed points of semi-groups of endomorphisms of linear spaces,Proc. Fifth Berkeley Symp. Math. Stat. Prob. 1966,II(1) 55–61.Google Scholar
  29. 29.
    Tempelman, A. (1986).Ergodic Theorems on Groups (in Russian), Mokslas, Vilnius.Google Scholar
  30. 30.
    Bingham, N. H., and Goldie, C. M. (1982). Probabilistic and deterministic averaging,Trans. Amer. Math. Soc. 269, 453–480.Google Scholar
  31. 31.
    Stam, A. J. (1968). Laws of large numbers for functions of random walks with positive drift,Composition Math. 19, 299–333.Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Yves Derriennic
    • 1
  • Michael Lin
    • 2
  1. 1.Université de Bretagne OccidentaleBrestFrance
  2. 2.Ben-Gurion University of the NegevBeer-ShevaIsrael

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