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Deviation inequalities for contractive infinite memory processes

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  • Doukhan, Paul
  • Fan, Xiequan

Abstract

In this paper, we introduce a class of stochastic processes that encompasses many natural and widely used examples. A key feature of these processes is their infinite memory, which enables them to retain information from arbitrarily distant past states. Using the martingale decomposition method, we derive deviation and moment inequalities for separately Lipschitz functionals of such processes, under various moment conditions on certain dominating random variables. Our results extend those obtained for Markov chains by Dedecker and Fan [Stochastic Process. Appl., 2015], as well as recent results by Chazottes et al. [Ann. Appl. Probab., 2023] concerning specific infinite-memory models with sub-Gaussian concentration bounds. We also discuss an application to the stochastic gradient Langevin dynamics algorithm.

Suggested Citation

  • Doukhan, Paul & Fan, Xiequan, 2026. "Deviation inequalities for contractive infinite memory processes," Stochastic Processes and their Applications, Elsevier, vol. 191(C).
    • Handle: RePEc:eee:spapps:v:191:y:2026:i:c:s0304414925002224
    DOI: 10.1016/j.spa.2025.104778
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    References listed on IDEAS

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    1. Doukhan, Paul & Wintenberger, Olivier, 2008. "Weakly dependent chains with infinite memory," Stochastic Processes and their Applications, Elsevier, vol. 118(11), pages 1997-2013, 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. 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.
    4. Lesigne, Emmanuel & Volný, Dalibor, 2001. "Large deviations for martingales," Stochastic Processes and their Applications, Elsevier, vol. 96(1), pages 143-159, November.
    5. 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.
    6. Fan, Xiequan & Grama, Ion & Liu, Quansheng, 2012. "Hoeffding’s inequality for supermartingales," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3545-3559.
    7. Lanzinger, H. & Stadtmüller, U., 2000. "Maxima of increments of partial sums for certain subexponential distributions," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 307-322, April.
    8. Emmanuel Rio, 2009. "Moment Inequalities for Sums of Dependent Random Variables under Projective Conditions," Journal of Theoretical Probability, Springer, vol. 22(1), pages 146-163, March.
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