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 n↓0, 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
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},
     pages = {209--218},
     year = {2003},
     publisher = {EDP Sciences},
     volume = {7},
     doi = {10.1051/ps:2003005},
     mrnumber = {1956079},
     zbl = {1019.60021},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps:2003005/}
}
TY  - JOUR
AU  - Eichelsbacher, Peter
AU  - Löwe, Matthias
TI  - Moderate deviations for I.I.D. random variables
JO  - ESAIM: Probability and Statistics
PY  - 2003
SP  - 209
EP  - 218
VL  - 7
PB  - EDP Sciences
UR  - https://www.numdam.org/articles/10.1051/ps:2003005/
DO  - 10.1051/ps:2003005
LA  - en
ID  - PS_2003__7__209_0
ER  - 
%0 Journal Article
%A Eichelsbacher, Peter
%A Löwe, Matthias
%T Moderate deviations for I.I.D. random variables
%J ESAIM: Probability and Statistics
%D 2003
%P 209-218
%V 7
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/ps:2003005/
%R 10.1051/ps:2003005
%G en
%F PS_2003__7__209_0
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

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

[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. | Zbl | MR

[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). | Zbl | MR

[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). | Zbl | MR

[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. | Zbl | MR | Numdam

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

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

Cité par Sources :