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Orlicz Integrability of Additive Functionals of Harris Ergodic Markov Chains

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High Dimensional Probability VII

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

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Abstract

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).

Mathematics Subject Classification (2010). Primary 60J05, 60E15; Secondary 60K05, 60F05

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Acknowledgements

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

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Authors and Affiliations

  1. Institute of Mathematics, University of Warsaw, Banacha 2, 02–097, Warszawa, Poland

    Radosław Adamczak & Witold Bednorz

Authors
  1. Radosław Adamczak
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  2. Witold Bednorz
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Correspondence to Radosław Adamczak .

Editor information

Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, Georgia, USA

    Christian Houdré

  2. University of Delaware, Department of Applied Economics and Stat University of Delaware, Newark, Delaware, USA

    David M. Mason

  3. Centre national de la recherche scientifique, Laboratoire J.A. Dieudonné, Université Côte d’Azur, Nice, France

    Patricia Reynaud-Bouret

  4. Department of Mathematics, University of Tennessee Department of Mathematics, Knoxville, Tennessee, USA

    Jan Rosiński

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Adamczak, R., Bednorz, W. (2016). Orlicz Integrability of Additive Functionals of Harris Ergodic Markov Chains. In: Houdré, C., Mason, D., Reynaud-Bouret, P., Rosiński, J. (eds) High Dimensional Probability VII. Progress in Probability, vol 71. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-40519-3_13

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