The large deviation principle for certain series

Miguel A. Arcones

ESAIM: Probability and Statistics (2004)

  • Volume: 8, page 200-220
  • ISSN: 1292-8100

Abstract

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We study the large deviation principle for stochastic processes of the form { ∑ k = 1 ∞ x k ( t ) ξ k : t ∈ T } , where { ξ k } k = 1 ∞ is a sequence of i.i.d.r.v.’s with mean zero and x k ( t ) ∈ ℝ . We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition, we derive new concentration inequalities, which are of independent interest.

How to cite

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Arcones, Miguel A.. "The large deviation principle for certain series." ESAIM: Probability and Statistics 8 (2004): 200-220. <http://eudml.org/doc/245701>.

@article{Arcones2004,
abstract = {We study the large deviation principle for stochastic processes of the form $\lbrace \sum _\{k=1\}^\{\infty \}x_\{k\}(t)\xi _\{k\}:t\in T\rbrace $, where $\lbrace \xi _\{k\}\rbrace _\{k=1\}^\{\infty \}$ is a sequence of i.i.d.r.v.’s with mean zero and $x_\{k\}(t)\in \mathbb \{R\}$. We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition, we derive new concentration inequalities, which are of independent interest.},
author = {Arcones, Miguel A.},
journal = {ESAIM: Probability and Statistics},
keywords = {large deviations; stochastic processes; Large deviations},
language = {eng},
pages = {200-220},
publisher = {EDP-Sciences},
title = {The large deviation principle for certain series},
url = {http://eudml.org/doc/245701},
volume = {8},
year = {2004},
}

TY - JOUR
AU - Arcones, Miguel A.
TI - The large deviation principle for certain series
JO - ESAIM: Probability and Statistics
PY - 2004
PB - EDP-Sciences
VL - 8
SP - 200
EP - 220
AB - We study the large deviation principle for stochastic processes of the form $\lbrace \sum _{k=1}^{\infty }x_{k}(t)\xi _{k}:t\in T\rbrace $, where $\lbrace \xi _{k}\rbrace _{k=1}^{\infty }$ is a sequence of i.i.d.r.v.’s with mean zero and $x_{k}(t)\in \mathbb {R}$. We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition, we derive new concentration inequalities, which are of independent interest.
LA - eng
KW - large deviations; stochastic processes; Large deviations
UR - http://eudml.org/doc/245701
ER -

References

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  13. [13] M. Ledoux, The Concentration of Measure Phenomenon. American Mathematical Society, Providence, Rhode Island (2001). Zbl0995.60002MR1849347
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  15. [15] M. Talagrand, A new isoperimetric inequality and the concentration of measure phenomenon. Geometric aspects of functional analysis (1989–90), Springer, Berlin. Lect. Notes Math. 1469 (1991) 94–124. Zbl0818.46047
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