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Deviation Inequalities for Centered Additive Functionals of Recurrent Harris Processes Having General State Space

Author

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  • Eva Löcherbach

    (Université de Cergy-Pontoise)

  • Dasha Loukianova

    (Université d’Evry-Val d’Essonne)

Abstract

Let X be a Harris recurrent strong Markov process in continuous time with general Polish state space E, having invariant measure μ. In this paper we use the regeneration method to derive non asymptotic deviation bounds for $$P_{x}\biggl(\biggl|\int_0^tf(X_s)\,ds\biggr|\geq t^{\frac{1}{2}+\eta}\varepsilon \biggr)$$ in the positive recurrent case, for nice functions f with μ(f)=0 (f must be a charge). We generalize these bounds to the fully null-recurrent case in the moderate deviations regime. We obtain a Gaussian concentration bound for all functions f which are a charge. The rate of convergence is expressed in terms of the deterministic equivalent of the process. The main ingredient of the proof is Nummelin splitting in continuous time, which allows one to introduce regeneration times for the process on an enlarged state space.

Suggested Citation

  • Eva Löcherbach & Dasha Loukianova, 2012. "Deviation Inequalities for Centered Additive Functionals of Recurrent Harris Processes Having General State Space," Journal of Theoretical Probability, Springer, vol. 25(1), pages 231-261, March.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:1:d:10.1007_s10959-010-0310-y
    DOI: 10.1007/s10959-010-0310-y
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    References listed on IDEAS

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    1. Löcherbach, Eva & Loukianova, Dasha, 2009. "The law of iterated logarithm for additive functionals and martingale additive functionals of Harris recurrent Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2312-2335, July.
    2. Löcherbach, Eva & Loukianova, Dasha, 2008. "On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 118(8), pages 1301-1321, August.
    3. Djellout, H. & Guillin, A., 2001. "Moderate deviations for Markov chains with atom," Stochastic Processes and their Applications, Elsevier, vol. 95(2), pages 203-217, October.
    4. D. Loukianova & O. Loukianov, 2008. "Deterministic equivalents of additive functionals of recurrent diffusions and drift estimation," Statistical Inference for Stochastic Processes, Springer, vol. 11(2), pages 107-121, June.
    Full references (including those not matched with items on IDEAS)

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