On delta-method of moments and probabilistic sums

We discuss a general framework for determining asymptotics of the expected value of random variables of the form f(X) in terms of a function f and central moments of the random variable X. This method may be used for approximation of entropy, inverse moments, and some statistics of discrete random variables useful in analysis of some randomized algorithms. Our approach is based on some variant of the Delta Method of Moments. We formulate a general result for an arbitrary distribution and next we show its specific extension to random variables which are sums of identically distributed independent random variables. Our method simpli files previous proofs of results of several authors and can be automated to a large extent. We apply our method to the binomial, negative binomial, Poisson and hypergeometric distribution. We extend the class of functions for which our method is applicable to some subclass of exponential functions and double exponential functions for some cases.