Large deviations for independent random variables – Application to Erdös-Renyi’s functional law of large numbers

Jamal Najim

ESAIM: Probability and Statistics (2005)

  • Volume: 9, page 116-142
  • ISSN: 1292-8100

Abstract

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A Large Deviation Principle (LDP) is proved for the family 1 n ∑ 1 n 𝐟 ( x i n ) · Z i n where the deterministic probability measure 1 n ∑ 1 n δ x i n converges weakly to a probability measure R and ( Z i n ) i ∈ ℕ are ℝ d -valued independent random variables whose distribution depends on x i n and satisfies the following exponential moments condition: sup i , n 𝔼 e α * | Z i n | < + ∞ forsome 0 < α * < + ∞ . In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result, we extend Erdös and Rényi’s functional law of large numbers.

How to cite

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Najim, Jamal. "Large deviations for independent random variables – Application to Erdös-Renyi’s functional law of large numbers." ESAIM: Probability and Statistics 9 (2005): 116-142. <http://eudml.org/doc/245785>.

@article{Najim2005,
abstract = {A Large Deviation Principle (LDP) is proved for the family $\frac\{1\}\{n\}\sum _1^n \mathbf \{f\}(x_i^n) \cdot Z^n_i$ where the deterministic probability measure $\frac\{1\}\{n\}\sum _1^n \delta _\{x_i^n\}$ converges weakly to a probability measure $R$ and $(Z^n_i)_\{i\in \mathbb \{N\}\}$ are $\mathbb \{R\}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition:\[\hspace*\{-56.9055pt\}\sup \_\{i,n\} \{\mathbb \{E\}\}\{\rm e\}^\{\alpha ^* |Z\_i^n|\}&lt; +\infty \quad \textrm \{for some\}\quad 0&lt;\alpha ^*&lt;+\infty .\]In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result, we extend Erdös and Rényi’s functional law of large numbers.},
author = {Najim, Jamal},
journal = {ESAIM: Probability and Statistics},
keywords = {large deviations; epigraphical convergence; Erdös-Rényi’s law of large numbers; Erdős-Rényi’s law of large numbers},
language = {eng},
pages = {116-142},
publisher = {EDP-Sciences},
title = {Large deviations for independent random variables – Application to Erdös-Renyi’s functional law of large numbers},
url = {http://eudml.org/doc/245785},
volume = {9},
year = {2005},
}

TY - JOUR
AU - Najim, Jamal
TI - Large deviations for independent random variables – Application to Erdös-Renyi’s functional law of large numbers
JO - ESAIM: Probability and Statistics
PY - 2005
PB - EDP-Sciences
VL - 9
SP - 116
EP - 142
AB - A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}\sum _1^n \mathbf {f}(x_i^n) \cdot Z^n_i$ where the deterministic probability measure $\frac{1}{n}\sum _1^n \delta _{x_i^n}$ converges weakly to a probability measure $R$ and $(Z^n_i)_{i\in \mathbb {N}}$ are $\mathbb {R}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition:\[\hspace*{-56.9055pt}\sup _{i,n} {\mathbb {E}}{\rm e}^{\alpha ^* |Z_i^n|}&lt; +\infty \quad \textrm {for some}\quad 0&lt;\alpha ^*&lt;+\infty .\]In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result, we extend Erdös and Rényi’s functional law of large numbers.
LA - eng
KW - large deviations; epigraphical convergence; Erdös-Rényi’s law of large numbers; Erdős-Rényi’s law of large numbers
UR - http://eudml.org/doc/245785
ER -

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