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Moderate Deviations for I.I.D. Random Variables

Published online by Cambridge University Press:  15 May 2003

Peter Eichelsbacher
Affiliation:
Ruhr-Universität Bochum, Fakultät für Mathematik, NA3/68, 44780 Bochum, Germany; peter.eichelsbacher@ruhr-uni-bochum.de.
Matthias Löwe
Affiliation:
Department of Mathematics, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands; loewe@sci.kun.nl.
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Abstract

We derive necessary and sufficient conditions for a sum of i.i.d. random variables $\sum_{i=1}^n X_i/b_n$ – where $\frac {b_n} n \downarrow 0$, but $\frac {b_n} {\sqrt n} \uparrow \infty$ – 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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2003

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References

de Acosta, A., Moderate deviations and associated Laplace approximations for sums of independent random vectors. Trans. Amer. Math. Soc. 329 (1992) 357-375. CrossRef
M.A. Arcones, The large deviation principle for empirical processes. Preprint (1999).
van den Berg, M., Bolthausen, E. and den Hollander, F., Moderate deviations for the volume of the Wiener sausage. Ann. Math. 153 (2001) 355-406. CrossRef
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.
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Springer, New York (1998).
M. Djellout, Moderate deviations for martingale differences and applications to Φ-mixing sequences. Stochastics and Stochastic Reports (to appear).
P. Eichelsbacher and U. Schmock, Rank-dependent moderate deviations for U-empirical measures in strong topologies(submitted).
E. Giné and V. de la Pe na, Decoupling: From dependence to independence. Springer-Verlag (1999).
Ledoux, M., Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi. Ann. Inst. H. Poincaré 28 (1992) 267-280.
M. Ledoux and M. Talagrand, Probability in Banach Spaces. Springer-Verlag, Berlin (1991).
Löwe, M. and Merkl, F., Moderate deviations for longest increasing subsequences: The upper tail. Comm. Pure Appl. Math. 54 (2001) 1488-1520. CrossRef