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

By Quansheng Liu, F. Watbled. 2008

82 citations recorded by the source corpus.

We first obtain exponential inequalities for martingales. Let be a sequence of martingale differences relative to a filtration and set Sn=X1+...+Xn. We prove that if for some [delta]>0,Q>=1, K>0 and all k, a.s., then for some constant c>0 (depending only on [delta],Q and K) and all x>0, , where c(x)=cx2 if x[set membership, variant]]0,1] and c(x)=cxQ if 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.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d4704e68202aad89cfe01734352690843a93eb8a

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Citations in the corpus

A Concentration Bound for Stochastic Approximation via Alekseev’s Formula (2015, result): “This section needs a generalization of a concentration result from [28], which we prove separately as Theorem A.”
Hoeffding’s inequality for supermartingales (2011, methodology): “Further, inequalities for the probabilities P (Xn > x) have been obtained by Prohorov [26], Hoeffding [17], Azuma [1], Steiger [28,29], Freedman [13], Nagaev [21], Haeusler [16], McDiarmid [20], Pinelis [23], Talagrand [30], De La Peña [10], Lesigne and Volný [18], Dzhaparidze and van Zanten [11], N”