Asymptotic behaviors with convergence rates of distributions of negative-binomial sums

Use your smartphone to scan this QR code and download this article 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 < α < 2, it is quite hard to estimate in the case of α ∈ (0,1) via the Zolotarev's probability metric. Mathematics Subject Classification 2010: 60G50; 60F05; 60E07.


INTRODUCTION
We follow the notations used in 1 . A random variable N r,p is said to have negative-binomial distribution with two parameters p ∈ (0, 1) andr ∈ N, if its probability mass function is given in form be a sequence of independent, identically distributed (i.i.d.) random variables, independent of N r,p . Then, the sum S N r,p = X 1 + X 2 + · · · + X N r,p is called negative-binomial sum. It is easily seen that when r = 1, the negative-binomial sum reduces to a geometric sum (see 2,3 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 1,[4][5][6][7][8][9]. 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 (p n /r) 1/α S N r,pn via Gnedenko's Transfer Theorem (see 10 for more details), where 1 < α < 2, r ∈ N, and p n = θ /n for any θ ∈ (0, 1) . Moreover, using Zolotarev's probability metric, the rate of convergence in weak limit theorem for normalized negative-binomial sum (p n /r) 1/α S N r,pn will be estimated. It is clear that corresponding results for normalized geometric sums of i.i.d. random variables will be concluded when r = 1. From now on, the symbols D → and = D denote the convergence in distribution and equality in distribution, respectively. The set of real numbers is denoted by R = (−∞, +∞) and we will denote by R = (1, 2, . . . }the set of natural numbers.

PRELIMINARIES
We denote by X the set of random variables defined on a probability space (Ω, A , P) and denote by C(R) the set of all real-valued, bounded, uniformly continuous functions defined on R with norm Moreover, for any m ∈ N, m < s ≤ m + 1 and β = s − m, let us set where f (k) is derivative function of order k of f . Then, the Zolotarev's probability metric will be recalled as follows Definition 1. (11)(12)(13). Let X,Y ∈ X. Zolotarev's probability metric on X between two random variables X and Y, is defined by Let m = 1 and s = 2, Zolotarev's probability metric of order 2 is defined by where X,Y ∈ X and We shall use following properties of Zolotarev's probability metric in the next sections (see [11][12][13] ). 1. Zolotarev's probability metric is d s an ideal metric of order s,, i.e., for any c ̸ = 0, we have and with Z is independent of X and Y, we get The following lemma states the most important property of Zolotarev's probability metric which will be used in proofs of our results.
Proof. For any x, y ∈ R and f ∈ D 2 ,by Mean Value Theorem we have where z is between x and y. Moreover, since f ∈ D 2 , one has Hence, we obtain following inequality Therefore, for all X,Y ∈ X, 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 Y is said to have symmetric stable distribution with two parameters α ∈ (0, 2] and A random variable ξ is said to have symmetric Linnik distribution with two parameters α ∈ (0, 2] and σ > 0 denoted by ξ ∼ Linnik (α, σ ) , if its characteristic function is given by (see 1 A random variable Λ is said to have Generalized Linnik distribution with three parameters α ∈ (0, 2] , and r ∈ N, denoted by Λ ∼ GLinnik (α, σ , r) , if its characteristic function is given as (see 1

MAIN RESULTS
From now on, let r ∈ N be a fixed natural number, p n = θ n for any, and n ≥ 1. We first prove the following theorem.
Based on the ideality of order s = 2 of Zolotarev's probability metric and according to Lemma 1, it follows that The proof is straightforward. Remark 1 . Since Y ∼ Stable (α, 1) , according to ( 14 , Corollary 5, page 305) then E |Y | < ∞. Moreover, based on the finiteness of E |X 1 | and ∥f ′ ∥, a weak limit theorem for normalized non-random sum will be stated from Theorem 1 as follows . Proposition 1 . Let N r,p n NB (r, p n ) . Then, Proof. Since N r,p n ∼ NB (r, p n ) , the characteristic function of N r,p n is given by Hence, the characteristic function of ( Letting n → ∞, 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 Proof. According to Proposition 1, we have where θ ∈ (0, 1) and the density function of Gamma random variable G is defined by Furthermore, by Theorem 1, one has ) whose characteristic function is given by On account of Gnedenko's Transfer Theorem (see 10 ), it follows that where Λ is a random variable whose characteristic function is defined by 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. Let { X j , j ≥ 1 } be a sequence of i.i.d. random variables with E (|X 1 |) = ρ ∈ (0, +∞) . Assume that N r,p n ∼ NB(r, p n ), independent of X j for all j ≥ 1. Then, be a sequence of independent, symmetric Linnik distributed random variables with parameters α ∈ (1, 2) and σ = 1,independent of N r,p n . Then, the characteristic function of sum ∑ Hence, the characteristic function of sum (p n /r) 1/α ∑ N r,pn j=1 ξ j will be defined as follows Thus, On account of ideality of Zolotarev's probability metric of order s = 2 it follows that Since ξ 1 Linnik (α, 1) with 1 < α < 2, by Proposition 4.3.18 in ( 1 , page 212), we have On account of Lemma 1, one has 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 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, etc. Especially, the method of probability metrics is more effective.