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Large deviations for martingales

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  • 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.

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  • 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
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    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.
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    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.
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    11. Boucher, Thomas R., 2016. "A note on martingale deviation bounds," Statistics & Probability Letters, Elsevier, vol. 111(C), pages 8-11.
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    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.
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