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

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:119

Article

Liu, Quansheng^1,2  Watbled, Frederique¹ 

[1] Univ Europeenne Bretagne, Univ Bretagne Sud, LMAM, F-56017 Vannes, France

[2] Changsha Univ Sci & Technol, Sch Math & Comp Sci, Changsha 410076, Hunan, Peoples R China

关键词: Martingale differences;    Supermartingales;    Large deviation inequality;    Exponential inequality;    Bernstein's inequality;    Hoeffding and Azuma's inequality;    Directed polymers;    Random environment;    Concentration inequality;    Free energy;    Convergence rate;    Multiplicative cascades;   

DOI  :  10.1016/j.spa.2009.05.001

来源: Elsevier

【 摘 要 】

We first obtain exponential inequalities for martingales. Let (X(k)) (1 <= k <= n) be a sequence of martingale differences relative to a filtration (F(k)) and set S(n) = X(1) + ... + X(n). We prove that if for some delta > 0, Q >= 1, K > 0 and all k, E vertical bar e(delta vertical bar Xk vertical bar Q)vertical bar F(k-1)vertical bar <= K a.s., then for some constant c > 0 (depending only on delta, Q and K) and all x > 0, P parallel to vertical bar S(n)vertical bar > nx vertical bar <= 2e(-nc(x)), where c(x) = cx(2) if x is an element of vertical bar 0, 1 vertical bar and c(x) = cx(Q) if x > 1; the converse also holds if (X(i)) 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 L(p)); 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. (C) 2009 Elsevier B.V. All rights reserved.

【 授权许可】

Free