Poisson approximation for expectations of unbounded functions of independent random variables

Abstract

Under minimal moment conditions complete asymptotic expansions are obtained for expectations of unbounded functions of a finite family of independent random variables in the Poissonian setting when the distributions of the random variables have large atoms at zero.

The median of the poisson distribution

“…Since there are many stochastic processes Z connecting X and Y , it seems that the efficiency of the method depends on the right choice of Z. The preceding ideas are close in spirit to the Lindeberg method considered by Borisov and Ruzankin [8] to deal with Poisson approximation of independent integer-valued random variables, as well as to the consideration of Markov birth and death processes to evaluate Stein's factors, as introduced by Xia [20] (see also Barbour and Xia [7] and Weinberg [19]). …”

Approximation Methods in Probability Theory

“…In this section, we give a brief outline of the discrete version of the Lindeberg method as it is presented in [27]. The primary field of application of the method is estimation of the expectation of unbounded functions.…”

Two limit theorems for Markov binomial distribution

“…This result was then extended to more general cases of independent lattice variables; see, for example, [2,4,19] and the references therein. Similar results hold for convolutions on a measurable Abelian group and sums of m-dependent random variables; see [5] and [15].…”