POISSON APPROXIMATION FOR RANDOM SUMS OF POISSON RANDOM VARIABLES

In this paper, we use the Stein-Chen method to determine a bound for the total variation distance between the distribution of random sums of independent Poisson random variables and an appropriate Poisson distribution. Two examples have been given to illustrate the result obtained. AMS Subject Classification: 62E17, 60F05, 60G05


Introduction
Let X 1 , X 2 , ... be a sequence of independent Poisson random variables, each with probability P (X i = k) = e −µ i µ k i k! , k ∈ N ∪ {0}, where µ i > 0. It is well-known that the sum S n = n i=1 X i has a Poisson distribution with mean µ n = n i=1 µ i , that is, S n = P µn is a Poisson random variable with mean λ n . Let N be a nonnegative integer-valued random variable and independent of the X i 's. Consider the random sums of N independent Poisson random variables, S N = N i=1 X i , it is observed that S N has a Poisson distribution with mean µ N when N is given. In this paper, we are interested to approximate the distribution of S N by a Poisson distribution with mean λ = E(µ N ), in terms of the total variation distance between the distributions of S N and P λ , The method for determining the desired result of this study is the well-known Stein-Chen method, which is described and used to determine the result in Sections 2 and 3, respectively. In Section 4, some examples are provided to illustrate applications of the result. Conclusion of this study is presented in the last section.

Method
Stein [3] introduced a powerful and general method for bounding the error in the normal approximation. This method was developed and applied in the setting of Poisson approximation by Chen [2]. Stein's equation for Poisson distribution with mean λ > 0 is, for given h, of the form For C x = {0, ..., x}, the solution f A of (2.1) is of the form and Barbour et al. [1] showed that

Result
The following theorem presents a bound for d T V (S N , P λ ), which is the desired result. Theorem 3.1. For λ = E(µ N ), then we have (2.2). Thus, applying (2.1), we can have which completes the proof.
If X i 's are identically distributed, then the following corollary is an immediately consequence of the Theorem 3.1.

Examples
This section gives two examples to illustrate the result in the case of X i 's to be identically distributed. Example 3.1. For n (n ∈ N) is fixed, let N be a random variable with probability function Therefore, E(N ) = 3n 2 and E|N − E(N )| = n 2 . Let µ 1 = µ 2 = · · · = µ, then λ = 3nµ 2 and we have

Conclusion
In this study, the Stein-Chen method was used to determine a bound for the total variation distance between the distribution of the random sums of independent Poisson random variables and a Poisson distribution. With this bound, it can be seen that the result gives a good approximation when all µ i are small.