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

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.