On Edgeworth expansions for dependency-neighborhoods chain structures and Stein's method

Abstract

Let W be the sum of dependent random variables, and h(x) be a function. This paper provides an Edgeworth expansion of an arbitrary ``length'' for %E{h(W)} in terms of certain characteristics of dependency, and of the smoothness of h and/or the distribution of W. The core of the class of dependency structures for which these characteristics are meaningful is the local dependency, but in fact, the class is essentially wider. The remainder is estimated in terms of Lyapunov's ratios. The proof is based on a Stein's method.