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Deviation inequalities for separately Lipschitz functionals of iterated random functions

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  • Dedecker, Jérôme
  • Fan, Xiequan

Abstract

We consider an X-valued Markov chain X1,X2,…,Xn belonging to a class of iterated random functions, which is “one-step contracting” with respect to some distance d on X. If f is any separately Lipschitz function with respect to d, we use a well known decomposition of Sn=f(X1,…,Xn)−E[f(X1,…,Xn)] into a sum of martingale differences dk with respect to the natural filtration Fk. We show that each difference dk is bounded by a random variable ηk independent of Fk−1. Using this very strong property, we obtain a large variety of deviation inequalities for Sn, which are governed by the distribution of the ηk’s. Finally, we give an application of these inequalities to the Wasserstein distance between the empirical measure and the invariant distribution of the chain.

Suggested Citation

  • Dedecker, Jérôme & Fan, Xiequan, 2015. "Deviation inequalities for separately Lipschitz functionals of iterated random functions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 60-90.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:1:p:60-90
    DOI: 10.1016/j.spa.2014.08.001
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    References listed on IDEAS

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    1. Lesigne, Emmanuel & Volný, Dalibor, 2001. "Large deviations for martingales," Stochastic Processes and their Applications, Elsevier, vol. 96(1), pages 143-159, November.
    2. Liu, Quansheng & Watbled, Frédérique, 2009. "Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3101-3132, October.
    3. Fan, Xiequan & Grama, Ion & Liu, Quansheng, 2012. "Hoeffding’s inequality for supermartingales," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3545-3559.
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    Cited by:

    1. Riekert, Adrian, 2022. "Convergence rates for empirical measures of Markov chains in dual and Wasserstein distances," Statistics & Probability Letters, Elsevier, vol. 189(C).
    2. Benjamin Poignard & Manabu Asai, 2023. "Estimation of high-dimensional vector autoregression via sparse precision matrix," The Econometrics Journal, Royal Economic Society, vol. 26(2), pages 307-326.
    3. Rémy Garnier, 2022. "Concurrent neural network: a model of competition between times series," Annals of Operations Research, Springer, vol. 313(2), pages 945-964, June.
    4. Fan, Xiequan & Alquier, Pierre & Doukhan, Paul, 2022. "Deviation inequalities for stochastic approximation by averaging," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 452-485.
    5. Alquier Pierre & Doukhan Paul & Fan Xiequan, 2019. "Exponential inequalities for nonstationary Markov chains," Dependence Modeling, De Gruyter, vol. 7(1), pages 150-168, January.

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