A Cramér Type Theorem for Weighted Random Variables

Abstract

A Large Deviation Principle (LDP) is proved for the family $(1/n)\sum_1^n f(x_i^n) Z_i$ where $(1/n)\sum_1^n \delta_{x_i^n}$ converges weakly to a probability measure on $R$ and $(Z_i)_{i\in N}$ are $R^d$-valued independent and identically distributed random variables having some exponential moments, i.e., $$E e^{a |Z|} \lt \infty$$ for some $0 \lt a \lt \infty$. The main improvement of this work is the relaxation of the steepness assumption concerning the cumulant generating function of the variables $(Z_i)_{i \in N}$. In fact, Gärtner-Ellis' theorem is no longer available in this situation. As an application, we derive a LDP for the family of empirical measures $(1/n) \sum_1^n Z_i \delta_{x_i^n}$. These measures are of interest in estimation theory (see Gamboa et al., Csiszar et al.), gas theory (see Ellis et al., van den Berg et al.), etc. We also derive LDPs for empirical processes in the spirit of Mogul'skii's theorem. Various examples illustrate the scope of our results.