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

Abstract: We first obtain exponential inequalities for martingales. Let be a sequence of martingale differences relative to a filtration and set . We prove that if for some , and all , a.s., then for some constant (depending only on and ) and all , , where if and if ; the converse also holds if are independent and identically distributed. This extends Bernstein’s inequality for and Hoeffding’s inequality for . 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 ); 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. [Copyright &y& Elsevier]