کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
1156090 958802 2009 32 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات ریاضیات (عمومی)
پیش نمایش صفحه اول مقاله
Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment
چکیده انگلیسی

We first obtain exponential inequalities for martingales. Let (Xk)(1≤k≤n) be a sequence of martingale differences relative to a filtration (Fk)(Fk) and set Sn=X1+⋯+XnSn=X1+⋯+Xn. We prove that if for some δ>0,Q≥1δ>0,Q≥1, K>0K>0 and all kk, E[eδ|Xk|Q|Fk−1]≤K a.s., then for some constant c>0c>0 (depending only on δ,Qδ,Q and KK) and all x>0x>0, P[|Sn|>nx]≤2e−nc(x), where c(x)=cx2c(x)=cx2 if x∈]0,1]x∈]0,1] and c(x)=cxQc(x)=cxQ if x>1x>1; the converse also holds if (Xi)(Xi) are independent and identically distributed. This extends Bernstein’s inequality for Q=1Q=1 and Hoeffding’s inequality for Q=2Q=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 LpLp); 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.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Stochastic Processes and their Applications - Volume 119, Issue 10, October 2009, Pages 3101–3132
نویسندگان
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