New Bounds on Poisson Approximation for Random Sums of Independent Binomial Random Variables

Abstract In this paper, we use the Stein-Chen method to obtain new bounds on Poisson approximation for random sums of independent binomial random variables. Some results related to sums of independent binomial distributed random variables are also investigated. The received results in the present study are more general and sharper than some known results.


Introduction
In recent times, Poisson approximation problem for random sums of discrete random variables has attracted the attention of mathematicians.Readers who are interested in this problem can refer to (Hung & Giang, 2016b), (Kongudomthrap & Chaidee, 2012), (Teerapabolarn, 2013), (Teerapabolarn, 2014b), (Vellaisamy & Upadhye, 2009) and (Yannaros, 1991) for more details.We need to recall some results concerning the bounds in Poisson approximation for random sums of discrete random variables.
Let Z 1 , Z 2 , . . .be a sequence of independent Bernoulli random variables, each with probability of success P(Z i = 1) = p i = 1 − P(Z i = 0), i = 1, 2, . .., and let N be a positive integer-valued random variable and independent of Z i 's.Let U λ * bound for the total variation distance between the distributions of V N and U λ * as follows, see (Yannaros, 1991): Let X 1 , X 2 , . . ., X n be n independently distributed binomial random variables, each with probabilities where p i ∈ (0, 1); Suppose that N is a positive integer-valued random variable and independent of X i 's.Let U λ be a Poisson random variable with mean λ, In 2014, Teerapabolarn used the Stein-Chen method to obtain a uniform bound for the total variation distance between the distribution functions of W N and U λ as follows, see (Teerapabolarn, 2014a): This paper is organized as follows.The second section is a brief introduction to Stein-Chen method.In section 3, we give main results of this paper, and conclusions of this study are presented in the last section.

Preliminaries
The Stein-Chen method has been dealt with in detail in many articles (the reader is referred to (Chen, 1975) and (Barbour, Holst & Janson, 1992) for fuller development).The Stein-Chen method can be summarized as follows.
Let us denote by F W n (A) the probability distribution function of a discrete random variable W n ∈ A and we will be denoted by the Poisson distribution function (λ n > 0), defined on the set A ⊆ Z + .The best known method for estimating is basing on the following arguments (see (Chen, 1975) for more details).
Assume that h is a bounded real-valued function defined on Z + and Consider the function f (.) which is a solution of the Stein's equation Setting Give h = h A and take the expectation of both sides of the equation (3), we have Thus, the problem of estimating ∆ can be reduced to that of estimating the difference of the expectations According to Barbour et al. (see (Barbour, Holst & Janson, 1992), for Before starting the main results in next section, we also need the following lemmas, which is directly obtained from (Barbour, Holst & Janson, 1992) and (Teerapabolarn & Wongkasem, 2007).
Lemma 3 Let U λ N and U λ denote a Poisson random variable with mean λ N and λ, respectively.Then, for A ⊆ Z + , the total variation distance between the distributions of U λ N and U λ satisfies the following inequality:

Main Results
The following lemma is established for proving the main results.
Lemma 4 Let X 1 , X 2 , . . .be a sequence of independent binomial distributed random variables.Setting W n = n ∑ i=1 X i and where f is a bounded real-valued function defined on Z + .
Proof.We have .
This finishes the proof.
The following theorems present non-uniform and uniform bounds for the distance between the distribution functions of W N and U λ , which are the expected results.

A Uniform Bound on Poisson Approximation for Random Sums of Independent Binomial Random Variables
Theorem 1 For A ⊆ Z + , we have Proof.Let f = f A be defined as in (5) and applying (4), we have Taking account of Lemma 4 and Lemma 1, it follows that Thus, Combining ( 8) with ( 9), gives From Lemma 3 and (10), it follows the fact that This finishes the proof.
Remark 1 For r 1 = r 2 = ... = r n = 1, we have a uniform bound on Poisson approximation for the random sums of independent Bernoulli random variables: Remark 2 Let us consider: Thus, the bounds in ( 7) and ( 11) are sharper than the bounds in ( 2) and (1), respectively.
Remark 3 The result ( 12) is a uniform bound on Poisson approximation for sums of independent binomial random variables.This bound is sharper than those reported in (Teerapabolarn, 2014a).

A Non-uniform Bound on Poisson Approximation for Random Sums of Independent Binomial Random Variables
Theorem 2 For w 0 ∈ Z + , we have Proof.For C w = {0, ..., w} and w 0 ∈ Z + , let h w 0 : According to Barbour et al. (see (Barbour, Holst & Janson, 1992) on p.7), the solution f C w 0 (w) of ( 3) is expressed in the form of Given f = f C w 0 and h = h C w 0 , the Stein's equation Taking expectations of both sides, and applying Lemma 2 and Lemma 4, we have Thus, In addition, by using Lemma 3, Teerapabolarn showed that (see (Teerapabolarn, 2013) for more details): Combining ( 14) and ( 15 This finishes the proof.