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A martingale inequality and large deviations

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  • Li, Yulin

Abstract

Let (Xi) be a martingale difference sequence and let Sn=[summation operator]i=1nXi. Suppose (Xi) is bounded in Lp. In the case p[greater-or-equal, slanted]2, Lesigne and Volný (Stochastic Process. Appl. 96 (2001) 143) obtained the estimation [mu](Sn>n)[less-than-or-equals, slant]cn-p/2, which is optimal in a certain sense. In this article, we show that [mu](Sn>n)[less-than-or-equals, slant]cn1-p when p[set membership, variant](1,2]. This is optimal for an i.i.d. sequence, as shown in Lesigne and Volný (Stochastic Process. Appl. 96 (2001) 143). For this purpose, we establish some inequalities for (Xi), which may be of interest on their own right.

Suggested Citation

  • Li, Yulin, 2003. "A martingale inequality and large deviations," Statistics & Probability Letters, Elsevier, vol. 62(3), pages 317-321, April.
    • Handle: RePEc:eee:stapro:v:62:y:2003:i:3:p:317-321
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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.
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    Cited by:

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    2. Meng, Yanjiao & Lin, Zhengyan, 2009. "On the weak laws of large numbers for arrays of random variables," Statistics & Probability Letters, Elsevier, vol. 79(23), pages 2405-2414, December.
    3. 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.
    4. Boucher, Thomas R., 2016. "A note on martingale deviation bounds," Statistics & Probability Letters, Elsevier, vol. 111(C), pages 8-11.
    5. Meng, Yanjiao & Lin, Zhengyan, 2009. "Maximal inequalities and laws of large numbers for Lq-mixingale arrays," Statistics & Probability Letters, Elsevier, vol. 79(13), pages 1539-1547, July.
    6. Li, Bainian & Zhang, Kongsheng & Wu, Libin, 2011. "A sharp inequality for martingales and its applications," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1260-1266, August.
    7. Eunji Lim, 2011. "On the Convergence Rate for Stochastic Approximation in the Nonsmooth Setting," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 527-537, August.

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