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Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment

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  • Liu, Quansheng
  • Watbled, Frédérique

Abstract

We first obtain exponential inequalities for martingales. Let be a sequence of martingale differences relative to a filtration and set Sn=X1+...+Xn. We prove that if for some [delta]>0,Q>=1, K>0 and all k, a.s., then for some constant c>0 (depending only on [delta],Q and K) and all x>0, , where c(x)=cx2 if x[set membership, variant]]0,1] and c(x)=cxQ if x>1; the converse also holds if (Xi) are independent and identically distributed. This extends Bernstein's inequality for Q=1 and Hoeffding's inequality for Q=2. We then apply the preceding result to establish exponential concentration inequalities for the free energy of directed polymers in a random environment and obtain upper bounds for its rates of convergence (in probability, almost surely and in Lp); we also give an expression for the free energy in terms of those of some multiplicative cascades, which improves an inequality of Comets and Vargas [Francis Comets, Vincent Vargas, Majorizing multiplicative cascades for directed polymers in random media, ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 267-277 (electronic)] to an equality.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:10:p:3101-3132
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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.
    2. Liu, Quansheng, 2000. "On generalized multiplicative cascades," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 263-286, April.
    3. Carmona, Philippe & Hu, Yueyun, 2004. "Fluctuation exponents and large deviations for directed polymers in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 285-308, August.
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    Cited by:

    1. 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.
    2. Martin Mbele Bidima & Miklos Rasonyi, 2012. "On long-term arbitrage opportunities in Markovian models of financial markets," Annals of Operations Research, Springer, vol. 200(1), pages 131-146, November.
    3. Francis Comets & Nobuo Yoshida, 2011. "Branching Random Walks in Space–Time Random Environment: Survival Probability, Global and Local Growth Rates," Journal of Theoretical Probability, Springer, vol. 24(3), pages 657-687, September.
    4. Fan, Xiequan & Grama, Ion & Liu, Quansheng, 2012. "Hoeffding’s inequality for supermartingales," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3545-3559.
    5. Shin-ichiro Takazawa, 2012. "Exponential inequalities and the law of the iterated logarithm in the unbounded forecasting game," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(3), pages 615-632, June.

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