Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment
Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment
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DOI：
10.1016/j.spa.2009.05.001
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发表时间：
2008-12
影响因子：
1.4
通讯作者：
Quansheng Liu;F. Watbled
中科院分区：
文献类型：
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作者：
Quansheng Liu;F. Watbled
首先得到了鞅的指数不等式。设（Xk）（1≤k≤n）是关于滤子（Fk）的鞅差序列，Sn=X1+ X2 +Xn.我们证明了，如果对某个δ>0，Q≥1，K>0和所有k，[公式：见正文] a.s.，则对于某个常数c>0（仅取决于δ，Q和K）和所有x>0，P[|SN|>nx]≤ 2 e −nc（x），其中c（x）= cx 2 if x∈] 0，1]且c（x）= cxQ if x>1;如果（Xi）是独立同分布的，则匡威亦然。这扩展了Q=1时的伯恩斯坦不等式和Q=2时的霍夫丁不等式。然后应用上述结果建立了随机环境中定向聚合物自由能的指数浓度不等式，并得到了其收敛速度的上界（在概率、几乎肯定和Lp中）;我们还给出了自由能的一个表达式，它改进了Comets和Vargas的一个不等式[弗朗西斯Comets，Vincent Vargas，随机介质中定向聚合物的优化乘法级联。Am. J. Probab.数学。统计。2（2006），267-277（电子）]到平等。
We first obtain exponential inequalities for martingales. Let (Xk)(1≤k≤n) be a sequence of martingale differences relative to a filtration (Fk) and set Sn=X1+⋯+Xn. We prove that if for some δ>0,Q≥1, K>0 and all k, [Formula: see text] a.s., then for some constant c>0 (depending only on δ,Q and K) and all x>0, P[|Sn|>nx]≤2e−nc(x), where c(x)=cx2if x∈]0,1] and c(x)=cxQif 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.