## Abstract

The negative-binomial sum is an extension of a geometric sum. It has been arisen from the necessity to resolve practical problems in telecommunications, network analysis, stochastic finance and insurance mathematics, etc. Up to the present, the topics related to negative-binomial sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned the rates of convergence for negative-binomial sums are still restrictive. The main purpose of this paper is to establish some weak limit theorems for negative-binomial sums of independent, identically distributed (i.i.d.) random variables via Gnedenko's Transfer Theorem originated by Gnedenko and Fahim (1969). Using Zolotarev's probability metric, the rate of convergence in weak limit theorems for negativebinomial sum are established. The received results are the rates of convergence in weak limit theorem for partial sum of i.i.d random variables related to symmetric stable distribution (Theorem 1), and asymptotic distribution together with the convergence rates for negative-binomial sums of i.i.d. random variables concerning to symmetric Linnik laws and Generalized Linnik distribution (Theorem 2 and Theorem 3). Based on the results of this paper, the analogous results for geometric sums of i.i.d. random variables will be concluded as direct consequences. However, the article has just been solved for the case of 1 <a < 2; it is quite hard to estimate in the case of a 2 (0;1) via the Zolotarev's probability metric.

**Mathematics Subject Classification 2010: 60G50; 60F05; 60E07.**

**Keywords:**Negative-binomial sum Gnedenko's Transfer Theorem Zolotarev's probability metric symmetric stable distribution symmetric Linnik distribution Generalized Linnik distribution

## Introduction

We follow the notations used in
1
. A random variable
is said to have
*
negative-binomial distribution with two parameters *
*
an*
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
*
negative-binomial sum*
. 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,
*
etc*
. 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
*
et al*
. (2018), Hung and Hau (2018),
*
etc*
. (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 Gnedenkoâ€™s Transfer Theorem (see 10 for more details), where and for any Moreover, using Zolotarevâ€™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 Zolotarevâ€™s probability metric will be recalled as follows

**
Definition 1. **
(
13
,
12
,
11
). Let
*
Zolotarevâ€™s probability metric*
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
,
*
i.e*
., 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. **
*
Let*
*
with *
*
and *
*
. The*
n

*
where *

**
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
*
symmetric stable distribution with two parameters*
*
and *
denoted by
if its characteristic function is given in form

A random variable
is said to have
*
symmetric Linnik distribution with two parameters *
*
and *
denoted by
if its characteristic function is given by (see
1
, page 199)

A random variable
is said to have G
*
eneralized Linnik distribution with three parameters
*
*
and *
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. **
*
Let *
*
be a sequence of i.i.d. random variables with *
*
Assume that *
*
and *
*
with *
*
Then*
,

**
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 1**
*
. *
*
Since *
*
according to (
14
, Corollary 5, page 305) then *
*
Moreover, based on the finiteness of *
*
and *
*
, a weak limit theorem for normalized non-random sum will be state*
d
*
from Theorem *
1
*
as follows*

**
Proposition 1**
*
. *
*
Let *
*
Then*
,

**
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 Gnedenkoâ€™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.**
*
Let
*
*
be a sequence of i.i.d. random variables with
*
*
Let *
*
independent o*
f
*
for all *
*
. *
*
Then,*

*
where *
*
with*

**
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 Gnedenkoâ€™s Transfer Theorem (see 10 ), 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 Zolotarevâ€™s probability metric by the following theorem.

**
Theorem 3.**
*
Let *
*
be a sequence of i.i.d. random variables with *
*
Assume that *
*
independent of *
*
for all *
*
. Then,*

*
where *
*
with *
*
*

**
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**
*
. *
*
From Theorem 1, a weak limit theorem for normalized negative-binomial random sum will be stated as follows*

## 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 Gnedenkoâ€™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 Gnedenkoâ€™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 Steinâ€™s method,
*
etc.*
Especially, the method of probability metrics is more effective. U sing the Zolotarevâ€™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 Zolotarevâ€™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, LĂ©vy-Prokhorov metric and probability metric based on Trotter operator,
*
etc*
.

## 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

## References

- Kotz S, Kozubowski T J, Podgorsky K. The Laplace Distribution and Generalization. Springer Science + Business Media, LLC; 2001.
- Hung T L. On the rate of convergence in limit theorems for geometric sums. Southeast-Asian J. of Sciences. 2013;2(2):117-130. Google Scholar
- Kien T L, . P T. The necessary and sufficient conditions for a probability distribution belongs to the domain of geometric attraction of standard Laplace distribution. Science & Technology Development Journal. 2019;22(1):143-146. Google Scholar
- Sunklodas J K. On the normal approximation of a negative binomial random sum. Lithuanian Mathematical Journal. 2015;55(1):150-158. Google Scholar
- Vellaisamy P, Upadhye N S. Compound Negative binomial approximations for sums of random variables. Probability and Mathematical Statistics. 2009;29(2):205-226. Google Scholar
- Sheeja S S, Kumar S. Negative binomial sum of random variables and modeling financial data. International Journal of Statistics and Applied Mathematics. 2017;2:44-51. Google Scholar
- Omair M, Almuhayfith F, Alzaid A. A bivariate model based on compound negative binomial distribution. Rev. Colombiana Estadist. 2018;41(1):87-108. Google Scholar
- Giang L T, Hung T L. An extension of random summations of independent and identically distributed random variables. Commun. Korean Math. Soc. 2018;33(2):605-618. Google Scholar
- Hung T L, Hau T N. On the accuracy of approximation of the distribution of negative-binomial random sums by the Gamma distribution. Kybernetika. 2018;54(5):921-936. Google Scholar
- Gnedenko B V, Fahim G. On a transfer theorem. Dokl. Akad. Nauk. SSSR. 1969;187(1):15-17. Google Scholar
- Zolotarev V. M.. Approximation of the distribution of sums of independent variables with values in infinite-dimensional spaces. Teor. Veroyatnost. i Primenen. 1976;21(4):741-758. Google Scholar
- Zolotarev V. M.. Probability metrics. Teor. Veroyatnost. i Primenen. 1983;28(2):264-287. Google Scholar
- Zolotarev V. M.. Metric distances in spaces of random variables and their distributions. Mat. Sb. (N.S.), Volume 101(143). 1976;3(11):416-454. Google Scholar
- Korolev V, Yu -. Product representations for random variables with Weibull distribution and their applications. Journal of Mathematical Sciences. 2016;218(3):29-313. Google Scholar
- Gennady S, Murad S T. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance.Boca Raton: Chapman & Hall/CRC; 1994.
- Hung T L. On a Probability Metric Based on Trotter Operator. Vietnam Journal of Mathematics. 2007;35:21-32. Google Scholar
- Yakumiv A L. Asymptotics at infinity of negative binomial infinitely divisible distributions. Theory Probab. Appl. 2011;55(2):342-351. Google Scholar