[Inégalités de grandes déviations pour les surmartingales et applications aux polymères dirigés en milieu aléatoire]
Nous montrons d'abord des inégalités exponentielles de grandes déviations pour les surmartingales, qui sont optimales dans le cas de variables aléatoires indépendantes et identiquement distribuées (iid). Nous utilisons ensuite ce résultat pour établir des inégalités exponentielles de concentration pour l'énergie libre des polymères dirigés en milieu aléatoire ; nous en déduisons des bornes supérieures pour sa vitesse de convergence (en probabilité, presque sûrement, et dans ), et une expression de l'énergie libre en terme d'énergies libres de certaines cascades multiplicatives.
We first show exponential large deviation inequalities for supermartingales, which are optimal in the independent and identically distributed (iid) case. We then apply them to establish new exponential concentration inequalities for the free energy of directed polymers in random environment; as consequences we obtain upper bounds of its rate of convergences (in probability, almost surely, and in ), and give an expression for the free energy in terms of free energies of some multiplicative cascades.
Accepté le :
Publié le :
@article{CRMATH_2009__347_19-20_1207_0,
author = {Liu, Quansheng and Watbled, Fr\'ed\'erique},
title = {Large deviation inequalities for supermartingales and applications to directed polymers in a random environment},
journal = {Comptes Rendus. Math\'ematique},
pages = {1207--1212},
publisher = {Elsevier},
volume = {347},
number = {19-20},
year = {2009},
doi = {10.1016/j.crma.2009.08.004},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.08.004/}
}
TY - JOUR AU - Liu, Quansheng AU - Watbled, Frédérique TI - Large deviation inequalities for supermartingales and applications to directed polymers in a random environment JO - Comptes Rendus. Mathématique PY - 2009 SP - 1207 EP - 1212 VL - 347 IS - 19-20 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.08.004/ DO - 10.1016/j.crma.2009.08.004 LA - en ID - CRMATH_2009__347_19-20_1207_0 ER -
%0 Journal Article %A Liu, Quansheng %A Watbled, Frédérique %T Large deviation inequalities for supermartingales and applications to directed polymers in a random environment %J Comptes Rendus. Mathématique %D 2009 %P 1207-1212 %V 347 %N 19-20 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.08.004/ %R 10.1016/j.crma.2009.08.004 %G en %F CRMATH_2009__347_19-20_1207_0
Liu, Quansheng; Watbled, Frédérique. Large deviation inequalities for supermartingales and applications to directed polymers in a random environment. Comptes Rendus. Mathématique, Tome 347 (2009) no. 19-20, pp. 1207-1212. doi : 10.1016/j.crma.2009.08.004. http://www.numdam.org/articles/10.1016/j.crma.2009.08.004/
[1] Fluctuation exponents and large deviations for directed polymers in a random environment, Stochastic Process. Appl., Volume 112 (2004) no. 2, pp. 285-308
[2] Directed polymers in a random environment: Path localization and strong disorder, Bernoulli, Volume 9 (2003) no. 4, pp. 705-723
[3] Majorizing multiplicative cascades for directed polymers in random media, ALEA Lat. Am. J. Probab. Math. Stat., Volume 2 (2006), pp. 267-277 (electronic)
[4] Large deviations for martingales, Stochastic Process. Appl., Volume 96 (2001) no. 1, pp. 143-159
[5] On generalized multiplicative cascades, Stochastic Process. Appl., Volume 86 (2000) no. 2, pp. 263-286
[6] Q. Liu, F. Watbled, Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment, Stochastic Process. Appl., in press
[7] Limit Theorems of Probability Theory, The Clarendon Press/Oxford University Press, New York, 1995
[8] Strong localization and macroscopic atoms for directed polymers, Probab. Theory Rel. Fields, Volume 138 (2007) no. 3–4, pp. 391-410
Cité par Sources :
