Compound Poisson Approximation Via Stein Method

Distribution approximation is an important portion of Probability and Statistics, and it’s also a important tool for the study and application in statistics. Stein method is an useful method which can be applied in distribution approximation. Stein method can be used to study the extreme quality of a random variable. Stein method can not only be effective when to prove constringency of a distribution, but also make some kind of probability distance between the random variable which we study and the corresponding approximation distribution be available.Stein method made its first appearance in an issue of Charles Stein about studying normal approximation in1970.Then Louise H.Y Chen applied Stein method to study poisson approximation, and this method is usually called Stein-Chen method. Stein’s method of obtaining rates of convergence is well known in normal and poisson approximation and have achieved much success. After Stein method’s first appearance, this method have been applied to many other distribution approximation areas in succession, such as multinomial approximation, gamma approximation, geometric approximation and so on. The aim of this paper is to extend Stein method to compound poisson distribution.In this paper, a bound of the total variation distance between the observed random variable W and a corresponding compound poisson distribution CP(λ) which is used to be the approximation distribution has be obtained by using the Stein equation for compound poisson approximation and studying the dependent relation between Ia,where W=Σα∈ΓVa, Va=ξaIa.and Ia are0-1random variables,{ξa,a, a∈Γ} are independent positive number-value random variables, Γ is a finite index set.