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

Published online by Cambridge University Press:  15 November 2005

Jamal Najim
Jamal Najim*
Affiliation:
CNRS, École Nationale Supérieure des Télécommunications, 46 rue Barrault 75634 Paris Cedex 13, France; [email protected]
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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: $$ \sup_{i,n} {\mathbb E}{\rm e}^{\alpha^* |Z_i^n|}< +\infty \quad\textrm{for some}\quad 0<\alpha^*<+\infty.$$

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.

Keywords

Large deviationsepigraphical convergenceErdös-Rényi's law of large numbers.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 9 , February 2005 , pp. 116 - 142
Copyright
© EDP Sciences, SMAI, 2005

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