Moderate deviations for I.I.D. random variables

Peter Eichelsbacher; Matthias Löwe

ESAIM: Probability and Statistics (2003)

  • 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 (2003): 209-218. <http://eudml.org/doc/244931>.

@article{Eichelsbacher2003,
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},
language = {eng},
pages = {209-218},
publisher = {EDP-Sciences},
title = {Moderate deviations for I.I.D. random variables},
url = {http://eudml.org/doc/244931},
volume = {7},
year = {2003},
}

TY - JOUR
AU - Eichelsbacher, Peter
AU - Löwe, Matthias
TI - Moderate deviations for I.I.D. random variables
JO - ESAIM: Probability and Statistics
PY - 2003
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
UR - http://eudml.org/doc/244931
ER -

References

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  7. [7] P. Eichelsbacher and U. Schmock, Rank-dependent moderate deviations for U -empirical measures in strong topologies (submitted). Zbl1039.60023
  8. [8] E. Giné and V. de la Peña, Decoupling: From dependence to independence. Springer-Verlag (1999). Zbl0918.60021MR1666908
  9. [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. Zbl0751.60009MR1162575
  10. [10] M. Ledoux and M. Talagrand, Probability in Banach Spaces. Springer-Verlag, Berlin (1991). Zbl0748.60004MR1102015
  11. [11] M. Löwe and F. Merkl, Moderate deviations for longest increasing subsequences: The upper tail. Comm. Pure Appl. Math. 54 (2001) 1488-1520. Zbl1033.60035MR1852980

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