Polynomial deviation bounds for recurrent Harris processes having general state space

2011/03/29 by Loecherbach, Eva, Loukianova, Dasha
#60F10 #60J35 #60J55 #62M05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1103.5610

Abstract

Consider a strong Markov process in continuous time, taking values in some Polish state space. Recently, Douc, Fort and Guillin (2009) introduced verifiable conditions in terms of a supermartingale property implying an explicit control of modulated moments of hitting times. We show how this control can be translated into a control of polynomial moments of abstract regeneration times which are obtained by using the regeneration method of Nummelin, extended to the time-continuous context. As a consequence, if a p-th moment of the regeneration times exists, we obtain non asymptotic deviation bounds of the form P_ν(|\frac1t∫₀^tf(X_s)ds-μ(f)|≥≥)≤ K(p)\frac1t^{p- 1}\frac 1≥^2(p-1)‖f‖_∞^2(p-1), p ≥ 2. Here, f is a bounded function and μ is the invariant measure of the process. We give several examples, including elliptic stochastic differential equations and stochastic differential equations driven by a jump noise.

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