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

Quansheng Liu and Frédérique Watbled

Stochastic Processes and their Applications, 2009, vol. 119, issue 10, 3101-3132

Abstract: 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.

Keywords: Martingale; differences; Supermartingales; Large; deviation; inequality; Exponential; inequality; Bernstein's; inequality; Hoeffding; and; Azuma's; inequality; Directed; polymers; Random; environment; Concentration; inequality; Free; energy; Convergence; rate; Multiplicative; cascades (search for similar items in EconPapers)
Date: 2009
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (6)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304-4149(09)00086-6
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:119:y:2009:i:10:p:3101-3132

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().