Moderate deviations for I.I.D. random variables
ESAIM: Probability and Statistics, Tome 7 (2003), pp. 209-218.

We derive necessary and sufficient conditions for a sum of i.i.d. random variables i=1 n X i /b n - where b n n0, but b n n - to satisfy a moderate deviations principle. Moreover we show that this equivalence is a typical moderate deviations phenomenon. It is not true in a large deviations regime.

DOI : 10.1051/ps:2003005
Classification : 60F10
Mots clés : moderate deviations, large deviations
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Eichelsbacher, Peter; Löwe, Matthias. Moderate deviations for I.I.D. random variables. ESAIM: Probability and Statistics, Tome 7 (2003), pp. 209-218. doi : 10.1051/ps:2003005. http://www.numdam.org/articles/10.1051/ps:2003005/

[1] A. De Acosta, Moderate deviations and associated Laplace approximations for sums of independent random vectors. Trans. Amer. Math. Soc. 329 (1992) 357-375. | MR | Zbl

[2] M.A. Arcones, The large deviation principle for empirical processes. Preprint (1999).

[3] M. Van Den Berg, E. Bolthausen and F. Den Hollander, Moderate deviations for the volume of the Wiener sausage. Ann. Math. 153 (2001) 355-406. | MR | Zbl

[4] H. Cramér, Sur un nouveau théorème-limite de la théorie des probabilités, Actualités Scientifique et Industrielles (736 Colloque consacré à la théorie des probabilités). Hermann (1938) 5-23. | JFM

[5] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Springer, New York (1998). | MR | Zbl

[6] M. Djellout, Moderate deviations for martingale differences and applications to φ-mixing sequences. Stochastics and Stochastic Reports (to appear). | Zbl

[7] P. Eichelsbacher and U. Schmock, Rank-dependent moderate deviations for U-empirical measures in strong topologies (submitted). | Zbl

[8] E. Giné and V. De La Peña, Decoupling: From dependence to independence. Springer-Verlag (1999). | MR | Zbl

[9] M. Ledoux, Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi. Ann. Inst. H. Poincaré 28 (1992) 267-280. | Numdam | MR | Zbl

[10] M. Ledoux and M. Talagrand, Probability in Banach Spaces. Springer-Verlag, Berlin (1991). | MR | Zbl

[11] M. Löwe and F. Merkl, Moderate deviations for longest increasing subsequences: The upper tail. Comm. Pure Appl. Math. 54 (2001) 1488-1520. | MR | Zbl

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