Asymptotic expansion in the limit theorem for the convergence to laplace distribution

Abstract

Let { ξ n } 1 ∞ be a sequence of non-negative identically distributed random variables with zero mean and finite variance σ 2, {ν = ν(ϵ)} a sequence for random variable independent of { ζ ν }, taking non-negative integer values and having a geometric distribution and with ζ ν = Σ 1 ν ζ ν . Gnedenko and Fraier. [Some notes on a work of Kowa lenka U. N. Letovsk Math. Sb. 1, 181–187 (1969)] proved a convergence theorem for ζ ν and some other authors proved other convergence theorems. This paper is concerned with some theorems about the asymptotic expansions and their consequences on the extremal properties of the Bernoulli random variable.