Moderate Deviations for I.I.D. Random Variables

Peter Eichelsbacher; Matthias Löwe

ESAIM: Probability and Statistics (2010)

  • Volume: 7, page 209-218
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Eichelsbacher, Peter, and Löwe, Matthias. "Moderate Deviations for I.I.D. Random Variables." ESAIM: Probability and Statistics 7 (2010): 209-218. <http://eudml.org/doc/104304>.

@article{Eichelsbacher2010,
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. },
author = {Eichelsbacher, Peter, Löwe, Matthias},
journal = {ESAIM: Probability and Statistics},
keywords = {Moderate deviations; large deviations.; moderate deviations; large deviations},
language = {eng},
month = {3},
pages = {209-218},
publisher = {EDP Sciences},
title = {Moderate Deviations for I.I.D. Random Variables},
url = {http://eudml.org/doc/104304},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Eichelsbacher, Peter
AU - Löwe, Matthias
TI - Moderate Deviations for I.I.D. Random Variables
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 7
SP - 209
EP - 218
AB - 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.
LA - eng
KW - Moderate deviations; large deviations.; moderate deviations; large deviations
UR - http://eudml.org/doc/104304
ER -

References

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  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.  Zbl1004.60021
  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.  Zbl64.0529.01
  5. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Springer, New York (1998).  Zbl0896.60013
  6. M. Djellout, Moderate deviations for martingale differences and applications to Φ-mixing sequences. Stochastics and Stochastic Reports (to appear).  Zbl1005.60044
  7. P. Eichelsbacher and U. Schmock, Rank-dependent moderate deviations for U-empirical measures in strong topologies(submitted).  Zbl1039.60023
  8. E. Giné and V. de la Pe na, Decoupling: From dependence to independence. Springer-Verlag (1999).  
  9. M. Ledoux, Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi. Ann. Inst. H. Poincaré28 (1992) 267-280.  Zbl0751.60009
  10. M. Ledoux and M. Talagrand, Probability in Banach Spaces. Springer-Verlag, Berlin (1991).  Zbl0748.60004
  11. M. Löwe and F. Merkl, Moderate deviations for longest increasing subsequences: The upper tail. Comm. Pure Appl. Math.54 (2001) 1488-1520.  Zbl1033.60035

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