Orlicz Integrability of Additive Functionals of Harris Ergodic Markov Chains

Conference paper
Part of the Progress in Probability book series (PRPR, volume 71)


For a Harris ergodic Markov chain (X n )n ≥ 0, on a general state space, started from the small measure or from the stationary distribution, we provide optimal estimates for Orlicz norms of sums i = 0 τ f(X i ), where τ is the first regeneration time of the chain. The estimates are expressed in terms of other Orlicz norms of the function f (with respect to the stationary distribution) and the regeneration time τ (with respect to the small measure). We provide applications to tail estimates for additive functionals of the chain (X n ) generated by unbounded functions as well as to classical limit theorems (CLT, LIL, Berry-Esseen).


Limit theorems Markov chains Orlicz spaces Tail inequalities Young functions 



Research partially supported by MNiSW Grant N N201 608740 and the Foundation for Polish Science.


  1. 1.
    R. Adamczak, A tail inequality for suprema of unbounded empirical processes with applications to Markov chains. Electron. J. Probab. 13 (34), 1000–1034 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    R. Adamczak, W. Bednorz, Exponential concentration inequalities for additive functionals of Markov chains. ESAIM: Probab. Stat. 19, 440–481 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    R. Adamczak, A.E. Litvak, A. Pajor, N. Tomczak-Jaegermann, Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling. Constr. Approx. 34 (1), 61–88 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    K.B. Athreya, P. Ney, A new approach to the limit theory of recurrent Markov chains. Trans. Am. Math. Soc. 245, 493–501 (1978)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    P.H. Baxendale, Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Probab. 15 (1B), 700–738 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    W. Bednorz, K. Łatuszyński, R. Latała, A regeneration proof of the central limit theorem for uniformly ergodic Markov chains. Electron. Commun. Probab. 13, 85–98 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    P. Bertail, S. Clémençon, Sharp bounds for the tails of functionals of Markov chains. Teor. Veroyatn. Primen. 54 (3), 609–619 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    E. Bolthausen, The Berry-Esseen theorem for functionals of discrete Markov chains. Z. Wahrsch. Verw. Gebiete 54 (1), 59–73 (1980)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    E. Bolthausen, The Berry-Esseén theorem for strongly mixing Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 60 (3), 283–289 (1982)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    R.C. Bradley, On quantiles and the central limit question for strongly mixing sequences. J. Theor. Probab. 10 (2), 507–555 (1997)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    R.C. Bradley, Jr., Information regularity and the central limit question. Rocky Mt. J. Math. 13 (1), 77–97 (1983)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    X. Chen, Limit theorems for functionals of ergodic Markov chains with general state space. Mem. Am. Math. Soc. 139 (664), xiv+203 (1999)Google Scholar
  13. 13.
    S.J.M. Clémençon, Moment and probability inequalities for sums of bounded additive functionals of regular Markov chains via the Nummelin splitting technique. Stat. Probab. Lett. 55 (3), 227–238 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    R. Douc, G. Fort, E. Moulines, P. Soulier, Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab. 14 (3), 1353–1377 (2004)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    R. Douc, A. Guillin, E. Moulines, Bounds on regeneration times and limit theorems for subgeometric Markov chains. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2), 239–257 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    P. Doukhan, P. Massart, E. Rio, The functional central limit theorem for strongly mixing processes. Ann. Inst. Henri Poincaré Probab. Stat. 30 (1), 63–82 (1994)MathSciNetMATHGoogle Scholar
  17. 17.
    U. Einmahl, D. Li, Characterization of LIL behavior in Banach space. Trans. Am. Math. Soc. 360 (12), 6677–6693 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    O. Häggström, On the central limit theorem for geometrically ergodic Markov chains. Probab. Theory Relat. Fields 132 (1), 74–82 (2005)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    P. Hitczenko, S.J. Montgomery-Smith, K. Oleszkiewicz, Moment inequalities for sums of certain independent symmetric random variables. Stud. Math. 123 (1), 15–42 (1997)MathSciNetMATHGoogle Scholar
  20. 20.
    G.L. Jones, On the Markov chain central limit theorem. Probab. Surv. 1, 299–320 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    I. Kontoyiannis, S.P. Meyn, Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13 (1), 304–362 (2003)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    M.A. Krasnoselskiĭ, J.B. Rutickiĭ, Convex Functions and Orlicz Spaces. Translated from the first Russian edition by Leo F. Boron (P. Noordhoff, Groningen, 1961)Google Scholar
  23. 23.
    K. Łatuszyński, B. Miasojedow, W. Niemiro, Nonasymptotic bounds on the mean square error for MCMC estimates via renewal techniques, in Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics and Statistics, vol. 23 (Springer, Heidelberg, 2012), pp. 539–555Google Scholar
  24. 24.
    K. Łatuszyński, B. Miasojedow, W. Niemiro, Nonasymptotic bounds on the estimation error of MCMC algorithms. Bernoulli 19 (5A), 2033–2066 (2014)MathSciNetMATHGoogle Scholar
  25. 25.
    L. Maligranda, Orlicz Spaces and Interpolation. Seminários de Matemática [Seminars in Mathematics], vol. 5 (Universidade Estadual de Campinas, Campinas, 1989)Google Scholar
  26. 26.
    L. Maligranda, E. Nakai, Pointwise multipliers of Orlicz spaces. Arch. Math. 95 (3), 251–256 (2010)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    F. Merlevède, M. Peligrad, E. Rio, A Bernstein type inequality and moderate deviations for weakly dependent sequences. Probab. Theory Relat. Fields 151 (3–4), 435–474 (2011)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    S. Meyn, R.L. Tweedie, Markov Chains and Stochastic Stability, 2nd edn. (Cambridge University Press, Cambridge, 2009)CrossRefMATHGoogle Scholar
  29. 29.
    S.J. Montgomery-Smith, Comparison of Orlicz-Lorentz spaces. Stud. Math. 103 (2), 161–189 (1992)MathSciNetMATHGoogle Scholar
  30. 30.
    E. Nummelin, A splitting technique for Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 43 (4), 309–318 (1978)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators. Cambridge Tracts in Mathematics, vol. 83 (Cambridge University Press, Cambridge, 1984)Google Scholar
  32. 32.
    E. Nummelin, P. Tuominen, Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory. Stoch. Process. Appl. 12 (2), 187–202 (1982)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    E. Nummelin, P. Tuominen, The rate of convergence in Orey’s theorem for Harris recurrent Markov chains with applications to renewal theory. Stoch. Process. Appl. 15 (3), 295–311 (1983)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    R. O’Neil, Fractional integration in Orlicz spaces. I. Trans. Am. Math. Soc., 115, 300–328 (1965)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    J.W. Pitman, An identity for stopping times of a Markov process, in Studies in Probability and Statistics (Papers in Honour of Edwin J. G. Pitman) (North-Holland, Amsterdam, 1976), pp. 41–57Google Scholar
  36. 36.
    J.W. Pitman, Occupation measures for Markov chains. Adv. Appl. Probab. 9 (1), 69–86 (1977)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146 (Marcel Dekker, New York, 1991)Google Scholar
  38. 38.
    E. Rio, The functional law of the iterated logarithm for stationary strongly mixing sequences. Ann. Probab. 23 (3), 1188–1203 (1995)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    E. Rio, Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiques & Applications, vol. 31 (Springer, Berlin, 2000)Google Scholar
  40. 40.
    G.O. Roberts, J.S. Rosenthal, General state space Markov chains and MCMC algorithms. Probab. Surv. 1, 20–71 (2004)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    S.M. Srivastava, A Course on Borel Sets. Graduate Texts in Mathematics, vol. 180 (Springer, New York, 1998)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WarsawWarszawaPoland

Personalised recommendations