ESAIM: Probability and Statistics (ESAIM: P&S)

ESAIM: P&S, April 2005, Vol. 9, pp. 116-142
DOI: 10.1051/ps:2005006

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

Jamal Najim

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

(Received November 13, 2003.)

Abstract
A Large Deviation Principle (LDP) is proved for the family $\frac\sum_1^n \mathbf{f}(x_i^n) \cdot Z^n_i$ where the deterministic probability measure $\frac\sum_1^n \delta_{x_i^n}$ converges weakly to a probability measure R and $(Z^n_i)_{i\in \mathbb$ are $\mathbb{R} ^d$ -valued independent random variables whose distribution depends on x_iⁿ and satisfies the following exponential moments condition:

$\begin{displaymath}\sup_{i,n} {\mathbb E}{\rm e}^{\alpha^* \vert Z_i^n\vert}< +\infty \quad\textrm{for some}\quad 0<\alpha^*<+\infty.\end{displaymath}$

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.

Mathematics Subject Classification. 46E30, 60F10, 60G57.

Key words: Large deviations, epigraphical convergence, Erdös-Rényi's law of large numbers.

© EDP Sciences, SMAI 2005