Large deviations for combinatorial distributions. I. Central limit theorems

Abstract

We prove a general central limit theorem for probabilities of large deviations for sequences of random variables satisfying certain natural analytic conditions. This theorem has wide applications to combinatorial structures and to the distribution of additive arithmetical functions. The method of proof is an extension of Kubilius’ version of Cram er’s classical method based on analytic moment generating functions. We thus generalize Cram er’s and Kubilius’ theorems on large deviations.