IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v96y2001i1p143-159.html
   My bibliography  Save this article

Large deviations for martingales

Author

Listed:
  • Lesigne, Emmanuel
  • Volný, Dalibor

Abstract

Let (Xi) be a martingale difference sequence and Sn=[summation operator]i=1n Xi. We prove that if supi E(eXi) 0 such that [mu](Sn>n)[less-than-or-equals, slant]e-cn1/3; this bound is optimal for the class of martingale difference sequences which are also strictly stationary and ergodic. If the sequence (Xi) is bounded in Lp, 2[less-than-or-equals, slant]p n)[less-than-or-equals, slant]cn-p/2 which is again optimal for strictly stationary and ergodic sequences of martingale differences. These estimations can be extended to martingale difference fields. The results are also compared with those for iid sequences; we give a simple proof that the estimate of Nagaev, Baum and Katz, [mu](Sn>n)=o(n1-p) for Xi[set membership, variant]Lp, 1[less-than-or-equals, slant]p 0.

Suggested Citation

  • Lesigne, Emmanuel & Volný, Dalibor, 2001. "Large deviations for martingales," Stochastic Processes and their Applications, Elsevier, vol. 96(1), pages 143-159, November.
    • Handle: RePEc:eee:spapps:v:96:y:2001:i:1:p:143-159
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(01)00112-0
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Volný, Dalibor, 1993. "Approximating martingales and the central limit theorem for strictly stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 44(1), pages 41-74, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Stoica, George, 2008. "The Baum-Katz theorem for bounded subsequences," Statistics & Probability Letters, Elsevier, vol. 78(7), pages 924-926, May.
    2. 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.
    3. Anders Bredahl Kock, 2013. "Oracle inequalities for high-dimensional panel data models," CREATES Research Papers 2013-20, Department of Economics and Business Economics, Aarhus University.
    4. Anders Bredahl Kock & Haihan Tang, 2014. "Inference in High-dimensional Dynamic Panel Data Models," CREATES Research Papers 2014-58, Department of Economics and Business Economics, Aarhus University.
    5. Lacoin, Hubert & Moreno, Gregorio, 2010. "Directed polymers on hierarchical lattices with site disorder," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 467-493, April.
    6. Giraudo, Davide, 2016. "Holderian weak invariance principle under a Hannan type condition," Stochastic Processes and their Applications, Elsevier, vol. 126(1), pages 290-311.
    7. Hua-Ming Wang, 2018. "Law of Large Numbers for Random Walk with Unbounded Jumps and Birth and Death Process with Bounded Jumps in Random Environment," Journal of Theoretical Probability, Springer, vol. 31(2), pages 619-642, June.
    8. H. Nooghabi & H. Azarnoosh, 2009. "Exponential inequality for negatively associated random variables," Statistical Papers, Springer, vol. 50(2), pages 419-428, March.
    9. Fan, Xiequan & Grama, Ion & Liu, Quansheng, 2012. "Hoeffding’s inequality for supermartingales," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3545-3559.
    10. Chen, Qisheng & Zhang, Qian & Liu, Chuan, 2019. "The pricing and numerical analysis of lookback options for mixed fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 123-128.
    11. Boucher, Thomas R., 2016. "A note on martingale deviation bounds," Statistics & Probability Letters, Elsevier, vol. 111(C), pages 8-11.
    12. 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.
    13. 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.
    14. 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.
    15. Kanaya, Shin & Otsu, Taisuke, 2012. "Large deviations of realized volatility," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 546-581.
    16. Oliveira, Paulo Eduardo, 2005. "An exponential inequality for associated variables," Statistics & Probability Letters, Elsevier, vol. 73(2), pages 189-197, June.
    17. Zhou, Xing-cai & Lin, Jin-guan, 2012. "A wavelet estimator in a nonparametric regression model with repeated measurements under martingale difference error’s structure," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1914-1922.
    18. Li, Yulin, 2003. "A martingale inequality and large deviations," Statistics & Probability Letters, Elsevier, vol. 62(3), pages 317-321, April.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wu, Wei Biao, 2009. "An asymptotic theory for sample covariances of Bernoulli shifts," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 453-467, February.
    2. Peligrad, Magda, 2020. "A new CLT for additive functionals of Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5695-5708.
    3. Jérôme Dedecker & Florence Merlevède & Dalibor Volný, 2007. "On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria," Journal of Theoretical Probability, Springer, vol. 20(4), pages 971-1004, December.
    4. Dalibor Volný, 2010. "Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem," Journal of Theoretical Probability, Springer, vol. 23(3), pages 888-903, September.
    5. Magda Peligrad & Hailin Sang, 2013. "Central Limit Theorem for Linear Processes with Infinite Variance," Journal of Theoretical Probability, Springer, vol. 26(1), pages 222-239, March.
    6. Yokoyama, Ryozo, 1995. "On the central limit theorem and law of the iterated logarithm for stationary processes with applications to linear processes," Stochastic Processes and their Applications, Elsevier, vol. 59(2), pages 343-351, October.
    7. Cuny, Christophe & Fan, Ai Hua, 2017. "Study of almost everywhere convergence of series by mean of martingale methods," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2725-2750.
    8. Dedecker, Jérôme & Merlevède, Florence, 2011. "Rates of convergence in the central limit theorem for linear statistics of martingale differences," Stochastic Processes and their Applications, Elsevier, vol. 121(5), pages 1013-1043, May.
    9. Dede, Sophie, 2009. "An empirical Central Limit Theorem in for stationary sequences," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3494-3515, October.
    10. Volný, Dalibor & Wang, Yizao, 2014. "An invariance principle for stationary random fields under Hannan’s condition," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4012-4029.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:96:y:2001:i:1:p:143-159. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.