Khinchin Families and Large Powers

The purpose of this paper is to present a wide picture of asymptotic results of large powers of coefficients of power series, with non negative coefficients, using as tools local central limit theorems for lattice random variables and for continuous families of lattice random variables combined with the theory of Khinchin families and of the Hayman class. We also cast in this framework the asymptotic formula due to Otter and to Meir-Moon of the coefficients of the solutions of Lagrange equations when the data power series has nonnegative coefficients. Asymptotics of Lagrangian probability distributions are also discussed.