Comptes Rendus
no. 10
Probability theory
A generalization of Cramér large deviations for martingales
[Une généralisation des grandes déviations de Cramér pour les martingales]

Présenté par : the Editorial Board

Xiequan Fan1, 2 ; Ion Grama1 ; Quansheng Liu1
1 Univ. Bretagne-Sud, UMR 6205, LMBA, F-56000 Vannes, France
2 Regularity Team, Inria and MAS Laboratory, École centrale de Paris, Grande Voie des Vignes, 92295 Châtenay-Malabry, France
Comptes Rendus. Mathématique, Volume 352 (2014) no. 10, pp. 853-858.

Dans cette note, nous donnons une généralisation des grandes déviations de Cramér pour les martingales, qui peut être considérée comme un supplément de Fan et al. (2013) [3]. Notre méthode est basée sur le changement de mesure de probabilité développé par Grama et Haeusler (2000) [6].

In this note, we give a generalization of Cramér's large deviations for martingales, which can be regarded as a supplement of Fan et al. (2013) [3]. Our method is based on the change of probability measure developed by Grama and Haeusler (2000) [6].

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.08.014
Xiequan Fan 1, 2 ; Ion Grama 1 ; Quansheng Liu 1

1 Univ. Bretagne-Sud, UMR 6205, LMBA, F-56000 Vannes, France
2 Regularity Team, Inria and MAS Laboratory, École centrale de Paris, Grande Voie des Vignes, 92295 Châtenay-Malabry, France 
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Xiequan Fan; Ion Grama; Quansheng Liu. A generalization of Cramér large deviations for martingales. Comptes Rendus. Mathématique, Volume 352 (2014) no. 10, pp. 853-858. doi : 10.1016/j.crma.2014.08.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.08.014/

[1] H. Cramér Sur un nouveau théorème-limite de la théorie des probabilités, Actual. Sci. Ind., Volume 736 (1938), pp. 5-23

[2] X. Fan; I. Grama; Q. Liu Sharp large deviations under Bernstein's condition, C. R. Acad. Sci. Paris, Sér. I, Volume 351 (2013), pp. 845-848

[3] X. Fan; I. Grama; Q. Liu Cramér large deviation expansions for martingales under Bernstein's condition, Stoch. Process. Appl., Volume 123 (2013), pp. 3919-3942

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[6] I. Grama; E. Haeusler Large deviations for martingales via Cramér's method, Stoch. Process. Appl., Volume 85 (2000), pp. 279-293

[7] I. Grama; E. Haeusler An asymptotic expansion for probabilities of moderate deviations for multivariate martingales, J. Theor. Probab., Volume 19 (2006), pp. 1-44

[8] E. Haeusler; K. Joos A nonuniform bound on the rate of convergence in the martingale central limit theorem, Ann. Probab., Volume 16 (1988) no. 4, pp. 1699-1720

[9] Q. Liu; F. Watbled Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment, Stoch. Process. Appl., Volume 119 (2009), pp. 3101-3132

[10] V.V. Petrov A generalization of Cramér's limit theorem, Usp. Mat. Nauk, Volume 9 (1954), pp. 195-202

[11] V.V. Petrov Sums of Independent Random Variables, Springer-Verlag, Berlin, 1975

[12] A. Račkauskas On probabilities of large deviations for martingales, Liet. Mat. Rink., Volume 30 (1990), pp. 784-795

[13] A. Račkauskas Large deviations for martingales with some applications, Acta Appl. Math., Volume 38 (1995), pp. 109-129

[14] A.I. Sakhanenko Berry–Esseen type bounds for large deviation probabilities, Sib. Math. J., Volume 32 (1991), pp. 647-656

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