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On bounds in Poisson approximation for integer-valued independent random variables
Journal of Inequalities and Applications volume 2014, Article number: 291 (2014)
The main aim of this note is to establish some bounds in Poisson approximation for row-wise arrays of independent integer-valued random variables via the Trotter-Renyi distance. Some results related to random sums of independent integer-valued random variables are also investigated.
MSC:60F05, 60G50, 41A36.
Let be a row-wise triangular array of independent integer-valued random variables with success probabilities ; ; ; ; ; . Set and . Suppose that (). We will denote by the Poisson random variable with mean λ. It has long been known that in the case of all (; ), the partial sum is said to be a Poisson-binomial random variable, and the probability distributions of , , are usually approximated by the distribution of . Specially, under the assumptions that , the well-known Poisson approximation theorem states that
Here, and from now on, the notation means the convergence in distribution. It is to be noticed that, for the information on the quality of the Poisson approximation, Le Cam (1960)  established the remarkable inequality
It is to be noticed that another inequality in Poisson approximation is usually expressed in terms of the total variation distance
where for the distributions P and Q on , the total variation distance between P and Q will be defined as follows:
In recent years many powerful tools for establishing the Le Cam’s inequality for a wide class of discrete independent random variables have been demonstrated, like the coupling technique, the Stein-Chen method, the semi-group method, the operator method, etc. Results of this nature may be found in [1–11], and .
The main aim of this paper is to establish the bounds of the Le Cam-style inequalities for independent discrete integer-valued random variables using the Trotter-Renyi distance based on Trotter-Renyi operator (see [13, 14], for more details). It is to be noticed that during the last several decades the Trotter-operator method has been used in many areas of probability theory and related fields. For a deeper discussion of Trotter’s operator we refer the reader to [12–20], and .
This paper is organized as follows. The second section deals with the definition and properties of Trotter-Renyi distance, based on Trotter’s operator and Renyi’s operator. Section 3 gives some results on Le Cam’s inequalities, based on the Trotter-Renyi distance, for independent integer-valued distributed random variables. The random versions of these results are also given in this section.
In the sequel we shall recall some properties of Trotter-Renyi operator, which has been used for a long time in various studies of classical limit theorems for sums of independent random variables (see [13–15, 18, 19], and , for the complete bibliography). Based on Renyi’s definition (, Chapter 8, Section 12, p.523), we redefine the Trotter-Renyi operator as follows.
Definition 2.1 The operator associated with a discrete random variable X is called the Trotter-Renyi operator, defined by
where by is denoted the class of all real-valued bounded functions f on the set of all non-negative integers . The norm of the function is defined by .
It is to be noticed that Renyi’s operator defined in  actually is a discrete form of Trotter’s operator (we refer the readers to [13, 15, 17–19], and , for a more general and detailed discussion of Trotter’s operator).
We shall need in the sequence the following main properties of Trotter-Renyi operator, for all functions and for :
Suppose that , are operators associated with two independent random variables X and Y. Then, for all ,
In fact, for all
Suppose that are the operators associated with the independent random variables . Then, for all , is the operator associated with the partial sum .
Suppose that and are operators associated with independent random variables and . Moreover, assume that all and are independent for . Then, for every ,(6)
Lemma 2.1 The equation for , shows that X and Y are identically distributed random variables.
Let be a sequence of Trotter-Renyi’s operators associated with the independent discrete random variables , and assume that is a Trotter-Renyi operator associated with the discrete random variable X. The following lemma states one of the most important properties of the Trotter-Renyi operator.
Lemma 2.2 A sufficient condition for a sequence of random variables to converge in distribution to a random variable X is that
Proof Since , for all , we conclude that
then we recover
It follows that as . We infer that as . □
Before stating the definition of the Trotter-Renyi distance we firstly need the definition of a probability metric. Let be a probability space and let be a space of real-valued -measurable random variables .
Definition 2.3 The Trotter-Renyi distance of two random variables X and Y with respect to the function is defined by
Based on the properties of the Trotter-Renyi operator, some properties of the Trotter-Renyi distance are summarized in the following (see [13, 14, 18, 19], and  for more details) and we shall omit the proofs.
It is easy to see that is a probability metric, i.e. for the random variables X, Y, and Z the following properties are possessed:
For every , the distance if .
for every .
for every .
If for every , then .
Let be a sequence of random variables and let X be a random variable. The condition
implies that as .
Suppose that ; are independent random variables (in each group). Then, for every ,(8)
Moreover, if the random variables are identically (in each group), then we have
Suppose that ; are independent random variables (in each group). Let be a sequence of positive integer-valued random variables that are independent of and . Then, for every ,(9)
Suppose that ; are independent identically distributed random variables (in each group). Let be a sequence of positive integer-valued random variables that are independent of and . Moreover, suppose that , . Then, for every , we have
Finally, we emphasize that the Trotter-Renyi distance in (7) and the total variation distance in (4) have a close relationship if the function f is chosen as an indicator function of a set , namely
where we denote by the total variation distance between two integer-valued random variables X and Y, defined as follows:
3 Main results
Let be a sequence of operators associated with the integer-valued random variables , ; , and let be a sequence of operators associated with the Poisson random variables with parameters , ; . Since is a Poisson random variable with positive parameter , we can write , where are independent Poisson random variables with positive parameters , and the notation denotes coincidence of distributions.
Theorem 3.1 Let be a row-wise triangular array of independent, integer-valued random variables with probabilities , ; ; ; ; . Let us write and . We will denote by the Poisson random variable with parameter . Then, for all functions ,
Proof Applying (8), we have
Moreover, for all , for all and we conclude that
Therefore, for all functions , and for all , we have
One infers that
Therefore, applying (8), we can assert that
This completes the proof. □
Corollary 3.1 Under the assumptions of Theorem 3.1, let , we have
Remark 3.1 We consider Corollary 3.1 and assume that the following conditions are satisfied:
Then as .
Theorem 3.2 Let be a row-wise triangular array of independent, integer-valued random variables with probabilities , ; ; ; ; . Moreover, we suppose that , are positive integer-valued random variables, independent of all , ; . Let us write and . We will denote by the Poisson random variable with parameter . Then, for all functions ,
Proof According to Theorem 3.1 and (9), for all functions , and for all , we have
The proof is complete. □
Corollary 3.2 According to Theorem 3.2, let , we have
Theorem 3.3 Let ( ; ) be a double array of independent integer-valued random variables with probabilities , , ; ; . Assume that for every the random variables , are independent, and for every the random variables are independent. Set . Let us denote by the Poisson random variable with mean . Then, for all ,
Proof Applying the inequality in (8), we have
According to Theorem 3.1, for all functions , and for all , we conclude that
This completes the proof. □
Theorem 3.4 Let be a double array of independent integer-valued random variables with ; ; ; ; ; . Assume that for every the random variables , are independent, and for every the random variables are independent. Set . Suppose that , are non-negative integer-valued random variables independent of all , ; . Let us denote by the Poisson random variable with mean . Then, for all functions ,
Proof According to Definition 2.1, we have
Therefore, for all functions , and for all , we have
The proof is straightforward. □
Remark 3.2 In the case of all probabilities , ; the partial sum will become a Poisson-binomial random variable, and one concludes that the results of Theorems 3.1, 3.2, 3.3, and 3.4 are extensions of results in  (see  for more details).
We conclude this paper with the following comments. The Trotter-Renyi distance method is based on the Trotter-Renyi operator and it has a big application in the Poisson approximation. Using this method it is possible to establish some bounds in the Poisson approximation for sums (or random sums) of independent integer-valued random vectors.
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The authors wish to express their gratitude to the referees for valuable remarks and comments, improving the previous version of this paper. The research was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED, Vietnam) under grant 101.01-2010.02.
The authors declare that they have no competing interests.
All authors contributed equally and significantly to this work. All authors drafted the manuscript, read and approved the final version of the manuscript.
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Hung, T.L., Giang, L.T. On bounds in Poisson approximation for integer-valued independent random variables. J Inequal Appl 2014, 291 (2014). https://doi.org/10.1186/1029-242X-2014-291
- Poisson approximation
- random sums
- Le Cam’s inequality
- Trotter’s operator
- Renyi’s operator
- probability distance
- integer-valued random variable