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

We derive necessary and sufficient conditions for a sum of i.i.d. random variables ${\sum }_{i=1}^{n}{X}_{i}/{b}_{n}$ - where $\frac{{b}_{n}}{n}↓0$, but $\frac{{b}_{n}}{\sqrt{n}}↑\infty$ - 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 : https://doi.org/10.1051/ps:2003005
Classification:  60F10
Keywords: moderate deviations, large deviations
@article{PS_2003__7__209_0,
author = {Eichelsbacher, Peter and L\"owe, Matthias},
title = {Moderate deviations for I.I.D. random variables},
journal = {ESAIM: Probability and Statistics},
publisher = {EDP-Sciences},
volume = {7},
year = {2003},
pages = {209-218},
doi = {10.1051/ps:2003005},
zbl = {1019.60021},
mrnumber = {1956079},
language = {en},
url = {http://www.numdam.org/item/PS_2003__7__209_0}
}

Eichelsbacher, Peter; Löwe, Matthias. Moderate deviations for I.I.D. random variables. ESAIM: Probability and Statistics, Volume 7 (2003) , pp. 209-218. doi : 10.1051/ps:2003005. http://www.numdam.org/item/PS_2003__7__209_0/

[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 1046015 | Zbl 0751.60007

[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 1829754 | Zbl 1004.60021

[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 64.0529.01

[5] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Springer, New York (1998). | MR 1619036 | Zbl 0896.60013

[6] M. Djellout, Moderate deviations for martingale differences and applications to $\phi$-mixing sequences. Stochastics and Stochastic Reports (to appear). | Zbl 1005.60044

[7] P. Eichelsbacher and U. Schmock, Rank-dependent moderate deviations for $U\phantom{\rule{-0.55542pt}{0ex}}$-empirical measures in strong topologies (submitted). | Zbl 1039.60023

[8] E. Giné and V. De La Peña, Decoupling: From dependence to independence. Springer-Verlag (1999). | MR 1666908 | Zbl 0918.60021

[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 1162575 | Zbl 0751.60009

[10] M. Ledoux and M. Talagrand, Probability in Banach Spaces. Springer-Verlag, Berlin (1991). | MR 1102015 | Zbl 0748.60004

[11] M. Löwe and F. Merkl, Moderate deviations for longest increasing subsequences: The upper tail. Comm. Pure Appl. Math. 54 (2001) 1488-1520. | MR 1852980 | Zbl 1033.60035