ESAIM: Probability and Statistics

Table of Contents - November 2005 - Volume 9  

Research Article

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

Najim, Jamal

CNRS, École Nationale Supérieure des Télécommunications, 46 rue Barrault 75634 Paris Cedex 13, France; najim@tsi.enst.fr

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.

(Received November 13 2003)

(Online publication November 15 2005)

Key Words:

  • Large deviations;
  • epigraphical convergence;
  • Erdös-Rényi's law of large numbers.

Mathematics Subject Classification:

  • 46E30;
  • 60F10;
  • 60G57
--