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 ( 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 δ > 0 , Q ≥ 1 , K > 0 and all k , E [ e δ | X k | Q | F k − 1 ] ≤ K a.s., then for some constant c > 0 (depending only on δ , Q and K ) and all x > 0 , P [ | S n | > n x ] ≤ 2 e − n c ( x ) , where c ( x ) = c x 2 if x ∈ ] 0 , 1 ] and c ( x ) = c x 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.