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Almost sure convergence of randomly truncated stochastic algorithms under verifiable conditions

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  • Lelong, Jérôme

Abstract

In this paper, we are interested in the almost sure convergence of randomly truncated stochastic algorithms. In their pioneering work Chen and Zhu [Chen, H., Zhu, Y., 1986. Stochastic Approximation Procedure with Randomly Varying Truncations. In: Scientia Sinica Series.] required that the family of the noise terms is summable to ensure the convergence. In our paper, we present a new convergence theorem which extends the already known results by making vanish this condition on the noise terms -- a condition which is quite hard to check in practice. The aim of this work is to prove an almost sure convergence result for randomly truncated stochastic algorithms under assumptions expressed independently of the algorithm paths so that the conditions can easily be verified in practical applications.

Suggested Citation

  • Lelong, Jérôme, 2008. "Almost sure convergence of randomly truncated stochastic algorithms under verifiable conditions," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2632-2636, November.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:16:p:2632-2636
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    Cited by:

    1. Devang Sinha & Siddhartha P. Chakrabarty, 2022. "Multilevel Monte Carlo and its Applications in Financial Engineering," Papers 2209.14549, arXiv.org.
    2. Frikha Noufel & Sagna Abass, 2012. "Quantization based recursive importance sampling," Monte Carlo Methods and Applications, De Gruyter, vol. 18(4), pages 287-326, December.
    3. Ahmed Kebaier & J'er^ome Lelong, 2015. "Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation," Papers 1510.03590, arXiv.org, revised Jul 2017.
    4. Ahmed Kebaier & Jérôme Lelong, 2017. "Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation," Working Papers hal-01214840, HAL.
    5. Teo Sharia, 2014. "Truncated stochastic approximation with moving bounds: convergence," Statistical Inference for Stochastic Processes, Springer, vol. 17(2), pages 163-179, July.
    6. Devang Sinha & Siddhartha P. Chakrabarty, 2022. "Multilevel Richardson-Romberg and Importance Sampling in Derivative Pricing," Papers 2209.00821, arXiv.org.
    7. Laetitia Badouraly Kassim & Jérôme Lelong & Imane Loumrhari, 2015. "Importance sampling for jump processes and applications to finance," Post-Print hal-00842362, HAL.
    8. Christophe Michel & Victor Reutenauer & Denis Talay & Etienne Tanr'e, 2015. "Liquidity costs: a new numerical methodology and an empirical study," Papers 1501.07404, arXiv.org, revised Dec 2015.
    9. Christophe Michel & Victor Reutenauer & Denis Talay & Etienne Tanré, 2016. "Liquidity costs: a new numerical methodology and an empirical study," Post-Print hal-01098096, HAL.
    10. repec:hal:wpaper:hal-00842362 is not listed on IDEAS
    11. Laetitia Badouraly Kassim & J'er^ome Lelong & Imane Loumrhari, 2013. "Importance sampling for jump processes and applications to finance," Papers 1307.2218, arXiv.org.
    12. Ahmed Kebaier & Jérôme Lelong, 2018. "Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation," Post-Print hal-01214840, HAL.
    13. Cénac P. & Maume-Deschamps V. & Prieur C., 2012. "Some multivariate risk indicators: Minimization by using a Kiefer–Wolfowitz approach to the mirror stochastic algorithm," Statistics & Risk Modeling, De Gruyter, vol. 29(1), pages 47-72, March.
    14. Lapeyre Bernard & Lelong Jérôme, 2011. "A framework for adaptive Monte Carlo procedures," Monte Carlo Methods and Applications, De Gruyter, vol. 17(1), pages 77-98, January.
    15. Ahmed Kebaier & Jérôme Lelong, 2018. "Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 611-641, June.

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