Introduction

We follow the notations used in1. A random variable is said to have d, if its probability mass function is given in form

Let be a sequence of independent, identically distributed (i.i.d.) random variables, independent of Then, the sum

is called . It is easily seen that when the negative-binomial sum reduces to a geometric sum (see 3, 2 and 1).

It is well-known that the topics related to negative-binomial sums have become the interesting research objects in probability theory. It has many applications in telecommunications, network analysis, stochastic finance and insurance mathematics, . Recently, problems concerning with negative-binomial sums have been investigated by Vellaisamy and Upadhye (2009), Yakumiv (2011), Sunklodas (2015), Sheeja and Kumar (2017), Giang and Hung (2018), Omair . (2018), Hung and Hau (2018), . (see 9, 1, 8, 7, 6, 5, 4). In many situations, some problems on the negative-binomial sums have not been fully studied yet, therefore its applications are still restrictive.

Therefore, the main aim of article is to establish weak limit theorems for normalized negative-binomial sums via Gnedenko’s Transfer Theorem (see 10 for more details), where and for any Moreover, using Zolotarev’s probability metric, the rate of convergence in weak limit theorem for normalized negative-binomial sum will be estimated. It is clear that corresponding results for normalized geometric sums of i.i.d. random variables will be concluded when

From now on, the symbols and denote the convergence in distribution and equality in distribution, respectively. The set of real numbers is denoted by and we will denote by the set of natural numbers.

PRELIMINARIES

We denote by the set of random variables defined on a probability space and denote by the set of all real-valued, bounded, uniformly continuous functions defined on with norm . Moreover, for any and let us set

and

where is derivative function of order of Then, the Zolotarev’s probability metric will be recalled as follows

Definition 1. (13, 12, 11). Let on between two random variables and is defined by

Let and Zolotarev's probability metric of order 2 is defined by

where and

We shall use following properties of Zolotarev's probability metric in the next sections (see 13, 12, 11).

1. Zolotarev's probability metric is an ideal metric of order , ., for any we have

and with is independent of and we get

2. If as then as

3. Let and be two independent sequences of i.i.d. random variables (in each sequence). Then, for all

The following lemma states the most important property of Zolotarev's probability metric which will be used in proofs of our results.

Lemma 1. n

Proof. For any and by Mean Value Theorem we have

where is between and Moreover, since one has

Hence, we obtain following inequality

Therefore, for all we get

The proof is straight-forward.

In the sequel, we shall recall several well-known distributions which are related to limit distributions of non-randomly sums and negative-binomial sums of i.i.d. random variables.

We follow the notations used in (1, page 204). A random variable is said to have denoted by if its characteristic function is given in form

A random variable is said to have denoted by if its characteristic function is given by (see 1, page 199)

A random variable is said to have G denoted by if its characteristic function is given as (see 1, page 216)

MAIN RESULTS

From now on, let be a fixed natural number, for any, and We first prove the following theorem.

Theorem 1. ,

Proof. Let be a sequence of i.i.d. copies of Then, it is clear that

Based on the ideality of order of Zolotarev's probability metric and according to Lemma 1, it follows that

The proof is straightforward.

Remark 1d1

Proposition 1,

Proof. Since the characteristic function of is given by

Hence, the characteristic function of is defined by

Letting we conclude that

This finishes the proof.

Using Gnedenko’s Transfer Theorem (see 10), a weak limit theorem for negative-binomial sum of i.i.d. random variables will be established as follows

Theorem 2.f

Proof. According to Proposition 1, we have

where and the density function of Gamma random variable is defined by

Furthermore, by Theorem 1, one has

where whose characteristic function is given by

On account of Gnedenko’s Transfer Theorem (see10), it follows that

where is a random variable whose characteristic function is defined by

Let us set then

The proof is immediate.

Next, the rate of convergence in Theorem 2 will be estimated via Zolotarev’s probability metric by the following theorem.

Theorem 3.

Proof. Let be a sequence of independent, symmetric Linnik distributed random variables with parameters and independent of . Then, the characteristic function of sum is defined by

Hence, the characteristic function of sum will be defined as follows

Thus,

On account of ideality of Zolotarev's probability metric of order it follows that

Since with by Proposition 4.3.18 in (1, page 212), we have

On account of Lemma 1, one has

Therefore,

The proof is complete.

Remark 2

Discussions

In some situations, it is quite hard to establish the limiting distributions for negative-binomial sums of i.i.d. random variables. Meanwhile, if the limiting distribution of the partial sum is stated, the limiting distribution of corresponding negative-binomial sum will be established by the Gnedenko’s Transfer Theorem (see 10). Thus, in this paper, the asymptotic behaviors of normalized negative-binomial random sums of i.i.d. random variables have been established via Gnedenko’s Transfer Theorem (Theorem 2).

Moreover, the mathematical tools have been used in study of convergence rates in limit theorems of probability theory including method of characteristic functions, method of linear operators, method of probability metrics and Stein’s method, Especially, the method of probability metrics is more effective. U sing the Zolotarev’s probability metric, the rates of convergence in weak limit theorem for partial sum and negative-binomial sum of i.i.d. random variables are estimated (Theorem 1 and Theorem 2).

It is worth pointing out that the Zolotarev’s probability metric used in this paper is an ideal metric, so it is easy to estimate approximations concerning random sums of random variables. Furthermore, this metric can be compared with well-known metrics like Kolmogorov metric, total variation metric, Lévy-Prokhorov metric and probability metric based on Trotter operator, .

Conclusions

A negative-binomial distributed random variable with two parameters and will reduce a geometric distributed random variable with parameter when. Hence, the analogous results for geometric sums of i.i.d. random variables will be concluded as direct consequences from this paper.

However, the article has just been solved for the case of it is very difficult to estimate in the case of via the Zolotarev's probability metric, but it will be considered in near future. Analogously, we can estimate the rates of convergence for negative-binomial sums of i.i.d. random variables for cases of and .

Competing interests

The authors declare that they have no competing interests.

Authors' Contributions

All authors contributed equally and significantly to this work. All authors drafted the manuscript, read and approved the final version of the manuscript.

Acknowledgments

The authors wish to express gratitude to mathematicians for sending their value published articles.

Uncited References

17, 16, 15

All authors contributed equally and significantly to this work. All authors drafted the manuscript, read and approved the final version of the manuscript.