On bounds in Poisson approximation for integer-valued independent random variables
© Hung and Giang; licensee Springer 2014
Received: 23 January 2014
Accepted: 10 July 2014
Published: 18 August 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.
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.
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).
- 5.Suppose that , are operators associated with two independent random variables X and Y. Then, for all ,
Suppose that are the operators associated with the independent random variables . Then, for all , is the operator associated with the partial sum .
- 7.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.
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 .
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 .
- 3.Let be a sequence of random variables and let X be a random variable. The condition
- 4.Suppose that ; are independent random variables (in each group). Then, for every ,(8)
- 5.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)
- 6.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
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.
This completes the proof. □
Then as .
The proof is complete. □
This completes the proof. □
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.
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.
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