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Concentration inequalities for additive functionals: A martingale approach

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  • Pepin, Bob

Abstract

This work shows how exponential concentration inequalities for additive functionals of stochastic processes over a finite time interval can be derived from concentration inequalities for martingales. The approach is entirely probabilistic and naturally includes time-inhomogeneous and non-stationary processes as well as initial laws concentrated on a single point. The class of processes studied includes martingales, Markov processes and general square integrable càdlàg processes. The general approach is complemented by a simple and direct method for martingales, diffusions and discrete-time Markov processes. The method is illustrated by deriving concentration inequalities for the Polyak–Ruppert algorithm, SDEs with time-dependent drift coefficients “contractive at infinity” with both Lipschitz and squared Lipschitz observables, some classical martingales and non-elliptic SDEs.

Suggested Citation

  • Pepin, Bob, 2021. "Concentration inequalities for additive functionals: A martingale approach," Stochastic Processes and their Applications, Elsevier, vol. 135(C), pages 103-138.
    • Handle: RePEc:eee:spapps:v:135:y:2021:i:c:p:103-138
    DOI: 10.1016/j.spa.2021.01.004
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    References listed on IDEAS

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    1. Guillin, Arnaud, 2001. "Moderate deviations of inhomogeneous functionals of Markov processes and application to averaging," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 287-313, April.
    2. Dzhaparidze, K. & van Zanten, J. H., 2001. "On Bernstein-type inequalities for martingales," Stochastic Processes and their Applications, Elsevier, vol. 93(1), pages 109-117, May.
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