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
Therefore,
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
Therefore,
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
and
Therefore, for all functions , and for all , we have
Thus,
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 [12] (see [12] 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.