Bounds for the approximation of Poisson-binomial distribution by Poisson distribution
© Hung and Thao; licensee Springer 2013
Received: 15 August 2012
Accepted: 9 January 2013
Published: 24 January 2013
Let (, ; ) be a row-wise triangular array of independent Bernoulli random variables with success probabilities , ; . For every , the random variables have probability distributions with complicated structure and therefore they are used to being approximated by Poisson distribution. Well-known Le Cam’s inequality is established for providing information on the quality of a Poisson approximation. The main aim of this paper is to re-establish the Le Cam-type inequalities via a linear operator. The operator method used in this paper is quite elementary and it also could be applied for the probability distributions of random sums in the Poisson approximation, where , , are positive integer-valued random variables, independent of all , ; .
MSC:60F05, 60G50, 41A36.
Throughout this paper, let (, ; ) be a row-wise triangular array of independent Bernoulli random variables with success probabilities , ; . The random variables , , are often called the Poisson-binomial random variables. And it is easily seen that the mean, variance, and characteristic function of , , are , , and , respectively.
It should be noted that in [6, 7], and  various powerful tools (such as the method of matrix analysis, the semi-group method, the coupling method, and the Chen-Stein method) for providing Le Cam’s inequality have been demonstrated. The main objective of this paper is to obtain the bounds for well-known Le Cam’s inequality in (3) using the operator method, introduced by Renyi . In the third section, we use the operator method from  to establish the bounds for the approximation of Poisson-binomial distribution by Poisson distribution. The operator method in this paper is quite elementary and it also could be applied for random sums , , where , are positive integer-valued random variables, independent of all , ; . This will be taken up in the last section. We refer the reader to the works of Trotter , Renyi , and Hung  for a deeper discussion of this operator method. Based on the operator method, the received results of this paper are analogues of Le Cam’s inequality in classical literature (we refer the reader to Steele , Le Cam , Chen , Neammanee , and Wang  for a complete treatment of the problem).
In the sequel we will need the operator method, which has been used for a long time in various studies of classical limit theorems for sums of independent random variables (see Trotter , Renyi , and Hung  for the complete bibliography).
We recall some definitions and notations. We denote by K the set of all real-valued bounded functions , defined on the set of non-negative integers . The norm of a function is defined by .
It is to be noticed that the linear operator defined in (4) is actually a discrete form of Trotter’s operator (we refer the readers to Trotter , Renyi , and Hung  for a more general and detailed discussion of this operator method).
We define the operator by , , and the product of two operators and is , .
for all .
for all .
Suppose that and are operators associated with two independent random variables X, Y and . Then .
Suppose that are the operators associated with the independent random variables . Then is the operator associated with the partial sum .
- 5.Suppose that and are operators associated with independent random variables and . Moreover, assume that all and are independent for . Then, for every ,(5)
It is to be noticed that .
- 7.Suppose that and are independent random variables (in each group), and let be a sequence of positive integer-valued random variables independent of all and , . Then, for every ,
Lemma 2.1 The equation for , , provided that X and Y are identically distributed random variables.
Let be a sequence of operators associated with the independent discrete random variables , and be the operator associated with the discrete random variable X. The following lemma states one of the most important properties of the operator .
It follows that as n tends to +∞.
In other words, as . □
3 A bound of Poisson-binomial approximation
Let , ; be the operators associated with the random variables , ; , and let , ; , be the operators associated with the Poisson random variables with parameters , ; . On the assumption that is a Poisson random variable with a positive parameter , we can perform that , where are independent Poisson random variables with positive parameters , and the notation denotes coincidence of distributions. We will now state an analogue of Le Cam’s inequality  via the linear operator in (4) as follows.
This completes the proof. □
The following corollaries are immediate consequences from Theorem 3.1.
The proof is complete. □
then the distribution of converges to the Poisson distribution with mean λ, i.e., as .
Thus, the proof is straightforward. □
4 A bound of random Poisson-binomial approximation
Here and subsequently, denotes the convergence in probability. For every , we denote by the random sums ( by convention). Therefore, the random sums could be said to be the random Poisson-binomial random variables. In this section, we establish Le Cam-type inequalities related to the Poisson approximation for distributions of random Poisson-binomial variables. It is to be noticed that many various results concerning the random summations have already been included in the textbooks of probability theory; see, e.g., [4, 13, 14]).
Let be operators associated with the independent triangular array of random variables , and let be operators associated with the independent Poisson distributed random variables with positive parameters . According to the properties of the linear operator in (4), we have and are the respective operators associated with the random sums and .
The proof is complete. □
Note that the following remarks are immediate consequences from Theorem 4.1.
When the success probability is identical, , ; for , we obtain the following remark.
It is worth noticing that when the positive integer-valued random variables , take on the value n with probability one, i.e., , the results concerning the probability distributions of the random sums in the Poisson approximation in this section return to the ones in Section 3.
We conclude this paper with the following comments. The linear operator in this paper introduced by Renyi  essentially is a discrete form of Trotter’s operator  which has been used in the theory of limit theorems. The proofs of theorems in this paper by the operator method are very elementary and elegant. The received results in this article allow us to think about a new approach method to the Poisson approximation problems for the distributions of the sums of the discrete independent random variables like Poisson-binomial, geometric, and negative binomial variables.
Dedicated to Professor Nguyen Duy Tien on the occasion of his 70th birthday.
The authors wish to express their gratitude to the referees for valuable remarks and comments improving the previous version of this paper. This work is supported by the Vietnam National Foundation For Science and Technology Development (NAFOSTED, Vietnam), grant 101.01-2010.02.
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