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

Science & Technology Development Journal, 22(4):415-421 Open Access Full Text Article Research Article 1University of Finance and Marketing, Vietnam 2DongThap Province, Vietnam Correspondence Phan Tri Kien, University of Finance and Marketing, Vietnam Email: phankien@ufm.edu.vn Asymptotic behaviors with convergence rates of distributions of negative-binomial sums Tran Loc Hung1, Phan Tri Kien1,*, Nguyen Tan Nhut2 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 neces- sity 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 mathemati- cians. 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.) ran- dom variables via Gnedenko's Transfer Theorem originated by Gnedenko and Fahim (1969). Us- ing Zolotarev's probability metric, the rate of convergence in weak limit theorems for negative- binomial sum are established. The received results are the rates of convergence in weak limit the- orem 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 Nr;p is said to have negative-binomial distribution with two parameters p 2 (0;1) andr 2 N, if its probability mass function is given in form P Nr;p = k
= k1 r1 ! pr(1 p)k1; k = r;r+1; : : : Let  X j; j  1 be a sequence of independent, identically distributed (i.i.d.) random variables, independent of Nr;p:Then, the sum SNr;p = X1+X2+


+XNr;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 and1). 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 1,4–9). In many situations, some problems on the negative-binomial sums have not been fully studied yet, therefore its applications are still restrictive. Cite this article : Loc Hung T, Tri Kien P, Tan Nhut N. Asymptotic behaviors with convergence rates of distributions of negative-binomial sums. Sci. Tech. Dev. J.; 22(4):415-421. 415 History  Received: 2019-06-20  Accepted: 2019-11-01  Published: 2019-12-31 DOI :10.32508/stdj.v22i4.1689 Copyright © VNU-HCM Press. This is an open- access article distributed under the terms of the Creative Commons Attribution 4.0 International license. Science & Technology Development Journal, 22(4):415-421 Therefore, the main aim of article is to establish weak limit theorems for normalized negative-binomial sums (pn=r) 1=aSNr;pn via Gnedenko’s Transfer Theorem (see 10 for more details), where 1< a < 2;r 2 N; and pn = q=n for any q 2 (0;1) :Moreover, using Zolotarev’s probability metric, the rate of convergence in weak limit theorem for normalized negative-binomial sum (pn=r)1=aSNr;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= Ddenote 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; : : :gthe set of natural numbers. PRELIMINARIES We denote by X the set of random variables defined on a probability space (W; A ; P) and denote byC(R) the set of all real-valued, bounded, uniformly continuous functions defined on R with norm k fk = sup x2R j f (x)j. Moreover, for any m 2 N; m < s m+1 and b = sm; let us set Cm(R) = n f 2C(R) : f (k) 2C(R); 1 k  m o and Ds = n f 2Cm(R) :  f (m)(x) f (m)(y) jx yjbo ; where f (k) is derivative function of order k of f :Then, the Zolotarev’s probability metric will be recalled as follows Definition 1. (11–13). Let X ;Y 2 X: Zolotarev’s probability metric on X between two random variables X and Y; is defined by dS(X ;Y ) = sup f2DS jE[ f (X) f (Y )]j: Let m= 1 and s= 2; Zolotarev’s probability metric of order 2 is defined by d2(X ;Y ) = sup f2D2 jE[ f (X) f (Y )]j; where X ;Y 2 X and D2 = n f 2C1(R) :  f 0(x) f 0(y) jx yjo : We shall use following properties of Zolotarev’s probability metric in the next sections (see 11–13). 1. Zolotarev’s probability metric is ds an ideal metric of order s;, i.e., for any c 6= 0; we have ds (cX ;cY ) = jcjsds (X ;Y ) ; and with Z is independent of X and Y; we get ds (X+Z;Y +Z) ds (X ;Y ) : 2. If ds (Xn;X0)! 0 as n! ¥; then Xn D! X0 as n! ¥: 3. Let  X j; j  1 and  Y j; j  1 be two independent sequences of i.i.d. random variables (in each sequence). Then, for all n 2 N; ds n å j=1 X j; n å j=1 Y j !  n:ds (X1;Y1) : The following lemma states the most important property of Zolotarev’s probability metric which will be used in proofs of our results. Lemma 1. LetX ;Y 2 X with E jX j< ¥ and E jY j< ¥. Then d2(X ;Y ) sup f2D2 f 0 :(EjX j+EjY j); 416 Science & Technology Development Journal, 22(4):415-421 where f 0 = s up w2R j f 0(w)j : Proof. For any x;y 2 R and f 2D2;by Mean ValueTheorem we have f (x) f (y) = (x y) f 0 (z) ; where z is between x and y:Moreover, since f 2D2; one has f 0(z) sup w2R  f 0(w) = f 0 : Hence, we obtain following inequality f (x) f (y) jx yj:  f 0(z) f 0 :jx yj: Therefore, for all X ;Y 2 X; we get d2(X;Y) = sup f2D2 jE[f(X) f(Y)]j  sup f2D2 f0
(EjXj+EjYj) . 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 a 2 (0;2] and s > 0; denoted by Y  Stable(a ;s) ; if its characteristic function is given in form jY (t) = exp sa jtja ; t 2 R: A random variable x is said to have symmetric Linnik distribution with two parameters a 2 (0;2] and s > 0 denoted by x  Linnik (a;s) ; if its characteristic function is given by (see1, page 199) jx (t) = 1 1+sa jtja ; t 2 R: A random variable L is said to have Generalized Linnik distribution with three parameters a 2 (0;2] ; and r 2 N; denoted by L GLinnik (a;s ;r) ; if its characteristic function is given as (see 1, page 216) jL (t) =  1 1+sa jtja r ; t 2 R: MAIN RESULTS From now on, let r 2 N be a fixed natural number, pn = qn for any, and n 1:We first prove the following theorem. Theorem 1. Let X j; j  1 be a sequence of i.i.d. random variables with E jX1j< ¥: Assume that S S table  a;(q=r)1=a  ; and Y  Stable(a;1) with 1< a < 2:Then, d2 (pn=r) 1=a n å j=1 X j;S !  n a2a (q=r) 2a
sup f2D2 f 0 :(E jX1j+EjY j) (3.1) Proof. Let Y j; j  1 be a sequence of i.i.d. copies of Y:Then, it is clear that S =D (pn=r) 1=a n å j=1 Y j: Based on the ideality of order s= 2 of Zolotarev’s probability metric and according to Lemma 1, it follows that 417 Science & Technology Development Journal, 22(4):415-421 d2 (pn=r) 1=a n å j=1 X j;S ! = d2 (pn=r) 1=a n å j=1 X j;(pn=r) 1=a n å j=1 Y j ! = (pn=r) 2=a d2 n å j=1 X j; n å j=1 Y j !  (pn=r)2=a
n
d2 (X1;Y1)  n a2a (q=r)2=a
sup f2D2 f0
(E jX1j+EjY j) The proof is straightforward. Remark 1 . Since Y  Stable(a ;1) ; according to (14, Corollary 5, page 305) then E jY j< ¥:Moreover, based on the finiteness of E jX1j and kf0k, a weak limit theorem for normalized non-random sum will be stated from Theorem 1 as follows (pn=r) 1=a n å j=1 X j D!S  Stable  a;(q=r)1=a  as n! ¥ (3.2) . Proposition 1 . Let Nr;pn NB(r; pn) :Then, Nr;pn n ! G Gamma(q ;r) ; as n! ¥: Proof. Since Nr;pn  NB(r; pn) ; the characteristic function of Nr;pn is given by jNr;pn (t) =  pneit 1 (1 pn)eit r ; t 2 R: Hence, the characteristic function of Nr;pnn is defined by j Nr;pn n (t) = j Nr;pn (t=n) = 1 1+ eit=n1
1pn !r = 0@ 1 1+  eit=n1 it=n  : itq 1Ar: Letting n! ¥; we conclude that lim n!¥j Nr;pnn (t) = lim n!¥ 0@ 1 1+  eit=n1 it=n  : itq 1Ar =  q q it r = jG (t) : 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. Let  X j; j  1 be a sequence of i.i.d. random variables with E(jX1j)< ¥: Let Nr;pn  NB(r; pn) ; independent of Xj for all j  1. Then, (pn=r)1=a Nr;pn å j=1 X j D! L as n ! ¥; where L GLinnik  a;r1=a ;r  with1< a < 2: Proof. According to Proposition 1, we have Nr;pn n ! G Gamma(q ;r) as n! ¥; 418 Science & Technology Development Journal, 22(4):415-421 where q 2 (0;1) and the density function of Gamma random variable G is defined by fG (x) = ( q r G(r)x r1eqx i f x> 0; 0 i f x 0: Furthermore, byTheorem 1, one has (pn=r)1=a n å j=1 X j D! S as n ! ¥; whereS  Stable  a;(q=r)1=a  whose characteristic function is given by jS (t) = exp (q=r) jtja ; t 2 R: On account of Gnedenko’s Transfer Theorem (see 10), it follows that (pn=r)1=a Nr;pn å j=1 X j D! L as n ! ¥; where L is a random variable whose characteristic function is defined by jL (t) = +¥Z 0 [jS (t)]w q r G(r) wr1eqwdw = q r G(r) +¥Z 0 e[qlnjS (t)]ww r1 dw: Let us set y= [q lnjS (t)]w; then jL(t) = q r G(r) +¥Z 0 eyyr1  1 q lnjs(t) r1 dy q lnjs(t) =  q q lnjs(t) r =  q q +(q=r)jtja r =  1 1+(1=r)jtja r The proof is immediate. Next, the rate of convergence inTheorem 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(jX1j) = r 2 (0;+¥) : Assume that Nr;pn  NB(r; pn); independent of X j for all j  1. Then, d2 (pn=r) 1=a Nrpn å j=1 X j;L !  n a2a
(q=r) 2aa sup f2D2 f 0
r+ 2 a sin pa  (3.3) , where L GLinnik  a;r1=a ;r  with 1< a < 2; and k f 0k= sup w2R j f 0(w)j : Proof. Let  x j; j  1 be a sequence of independent, symmetric Linnik distributed random variables with parameters a 2 (1;2) and s = 1;independent of Nr;pn . Then, the characteristic function of sum åNr;pnj=1 x jis defined by j å Nr;pn j=1 x j (t) = pnjx1 (t) 1 (1 pn)jx1 (t) !r =  pn pn+ jtja r : Hence, the characteristic function of sum (pn=r)1=aåNr;pnj=1 x j will be defined as follows j å Nr;pn j=1 x j h (pn=r) 1=a t i = 0B@ pn pn+  (pn=r) 1=a t a 1CA r = 1 1+ 1r jtja !r = jL(t): 419 Science & Technology Development Journal, 22(4):415-421 Thus, L=D(pn=r)1=a Nr;pn å j=1 x j: On account of ideality of Zolotarev’s probability metric of order s= 2 it follows that d2 (pn=r) 1=a Nr;pn å j=1 X j;L ! = d2 (pn=r) 1=a Nr;pn å j=1 X j;(pn=r) 1=a Nr;pn å j=1 x j ! = (pn=r) 2=a d2 Nr;pn å j=1 X j; Nr;pn å j=1 x j ! = = (pn=r) 2=a ¥ å q=1 ( P Nr;pn = q
d2 q å j=1 X j; q å j=1 x j !)  (pn=r)2=a ¥ å q=1  P Nr;pn = q
q
d2 (X1;x1) = (pn=r) 2=a E Nr;pn

d2 (X1;x1) Since x1 Linnik (a;1) with 1< a < 2; by Proposition 4.3.18 in (1, page 212), we have E(jx1j) = 2asin pa : On account of Lemma 1, one has d2 (X1;x1) sup f2D2 f0
(E jX1j+E jx1j) = sup f2D2 kf0k
 r+ 2 a sin pa  Therefore, d2 (pn=r) 1=a Nr;pn å j=1 X j;L !  (pn=r) 2a a sup f2D2 kf0k
 r+ 2 a sin pa  = n a2 a
(q=r) 2aa sup f2D2 kf0k
 r+ 2 a sin pa  The proof is complete. Remark 2 . FromTheorem 1, a weak limit theorem for normalized negative-binomial random sum will be stated as follows (pn=r) 1=a Nr;pn å j=1 X j D! L GLinnik  a;r1=a ;r  as n! ¥ (3.4) . 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. 420 Science & Technology Development Journal, 22(4):415-421 Using 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, etc. CONCLUSIONS A negative-binomial distributed random variable with two parameters r 2 N and pn 2 (0;1) will reduce a geometric distributed random variable with parameter pn 2 (0;1) whenr = 1. 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 1< a < 2; it is very difficult to estimate in the case of a 2 (0;1) 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 a = 1 and a = 2. 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. REFERENCES 1. Kotz S, Kozubowski TJ, Podgorsky K. Springer Science + Business Media, LLC; 2001. 2. Hung TL. On the rate of convergence in limit theorems for geometric sums. Southeast-Asian J of Sciences. 2013;2(2):117–130. 3. Kien TL, PT. 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. 4. Sunklodas JK. On the normal approximation of a negative binomial random sum. Lithuanian Mathematical Journal. 2015;55(1):150–158. 5. Vellaisamy P, Upadhye NS. Compound Negative binomial approximations for sums of random variables. Probability and Mathematical Statistics. 2009;29(2):205–226. 6. Sheeja SS, Kumar S. Negative binomial sum of random variables and modeling financial data. International Journal of Statistics and Applied Mathematics. 2017;2:44–51. 7. Omair M, Almuhayfith F, Alzaid A. A bivariate model based on compound negative binomial distribution. Rev Colombiana Estadist. 2018;41(1):87–108. 8. Giang LT, Hung TL. An extension of random summations of independent and identically distributed random variables. Commun Korean Math Soc. 2018;33(2):605–618. 9. Hung TL, Hau TN. On the accuracy of approximation of the distribution of negative-binomial random sums by the Gamma distribution. Kybernetika. 2018;54(5):921–936. 10. Gnedenko BV, Fahim G. On a transfer theorem. Dokl Akad Nauk SSSR. 1969;187(1):15–17. 11. Zolotarev VM. Approximation of the distribution of sums of independent variables with values in infinite-dimensional spaces. Teor Veroyatnost i Primenen. 1976;21(4):741–758. 12. Zolotarev VM. Probability metrics. Teor Veroyatnost i Primenen. 1983;28(2):264–287. 13. Zolotarev VM. Metric distances in spaces of random variables and their distributions. Mat Sb (NS), Volume 101(143). 1976;3(11):416–454. 14. Korolev V, Yu. Product representations for random variables with Weibull distribution and their applications. Journal of Mathematical Sciences. 2016;218(3):29–313. 421