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

Najim, Jamal 

ESAIM: Probability and Statistics, Tome 9 (2005), pp. 116-142

Résumé

A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}{\sum }_1^n𝐟\left(x_i^n\right)·Z_i^n$ where the deterministic probability measure $\frac{1}{n}{\sum }_1^n{\delta }_{x_i^n}$ converges weakly to a probability measure $R$ and ${\left(Z_i^n\right)}_{i\in \mathbb{N}}$ are ${ℝ}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition:

$$\underset{i,n}{sup}𝔼{\mathrm{e}}^{{\alpha }^{*}\left|Z_i^n\right|}<+\infty \mathrm{forsome}0<{\alpha }^{*}<+\infty .$$

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.

Détail

DOI : 10.1051/ps:2005006

Classification : 46E30, 60F10, 60G57
Keywords: large deviations, epigraphical convergence, Erdös-Rényi's law of large numbers