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

Abstract

A Large Deviation Principle (LDP) is proved for the family 1 n � n 1 f(x n ) · Z n where the de- terministic probability measure 1 n � n 1 δx n i converges weakly to a probability measure R and (Z n i )i∈N are R d -valued independent random variables whose distribution depends on x n and satisfies the following exponential moments condition: sup i,n Ee α ∗ |Zn i | < +∞ for some 0 <α ∗ < +∞. In this context, the identification of the rate function is non-trivial due to the absence of equidistribu- tion. We rely on fine convex analysis to address this issue. Among the applications of this result, we extend Erdos and Renyi's functional law of large numbers.