Complete Convergence for Weighted Sums of Weakly Negative Dependent of Random Variables

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1 Flomat 30:12 (2016), DOI /FIL N Publshed by Faculty of Scences and Mathematcs, Unversty of Nš, Serba Avalable at: Complete Convergence for Weghted Sums of Weakly Negatve Dependent of Random Varables H. Nader a, M. Amn a, A. Bozorgna a a Department of Statstcs, Faculty of Mathematcal Scence, Ferdows Unversty of Mashhad, Mashhad, Iran Abstract. Some basc theoretcal propertes and complete convergence theorems for weghted sums of weakly negatve dependent are provded and appled to random weghtng estmate. Moreover, varous examples are presented. 1. Introducton The concept of complete convergence ntroduced by Hsu and Robbns [6] as follows. The sequence {X, 1} of random varables convergence to constant C completely, f P( X n C > ε) < for every ε > 0. The complete convergence of dependent random varables has been nvestgated by, for example, Wang et al. [16], Ko [7], Wu [17], Sung [15], Jun and Deme [11], Amn et al. [1] and Deng et al. [4]. In ths paper, we derve some basc theoretcal propertes, complete convergence for weakly negatve dependent (WND) under condtons and appled the results to random weghtng estmate. The concept of WND random varables was ntroduced by Ranjbar et al. [12] n bvarate case, ths class s well defned and contans a large class of multnomal dstrbuton functons and some nterestng applcaton for WND sequence have been found. For example, for WND random varables wth heavy-tals Ranjbar et al. [13] obtaned the asymptotc behavour of product of two random varables and also, Ranjbar et al. [12] studed the asymptotc behavour of convoluton. The assumpton of WND for a sequence of random varables s specal case of extended negatve dependent (END), or negatve dependent (ND). So, t s very sgnfcant to study on a lmtng behavour of ths WND class. The paper s organzed as follows. Two mportant defntons, some basc propertes and the relatonshp between WND, END and ND are gven n Secton 2. The man results are presented n Secton 3 that applcaton of them are used about random weghted estmate. Further, n Secton 4 some well-known multvarate dstrbutons possess n Defnton Defntons and Prelmnares Ths secton contans man defntons and some propertes of WND random varables. Also, we wll provde some prelmnary facts needed for the proof of our man results n next secton Mathematcs Subject Classfcaton. Prmary 60F15 Keywords. Complete convergence, Weakly negatve dependent, Copula. Receved: 02 September 2014; Revsed: 09 March 2015; Accepted: 16 May 2015 Communcated by Mroslav M. Rstć Emal addresses: h nadersp@math.usb.ac.r (H. Nader), m-amn@um.ac.r (M. Amn), a.bozorgna@khayyam.ac.r (A. Bozorgna)

2 Nader et al. / Flomat 30:12 (2016), Defnton 2.1. (Ranjbar et al. [12]) The random vector X = (X 1,..., X n ) s sad to be weakly negatve dependent (WND) f there exsts a constant M 1 1, such that for n > 1 f (x 1,..., x n ) M 1 f (x ), where f (x 1,..., x n ) and f (x ) s are jont densty and margnal denstes of random vector X, respectvely. We say that f (x 1,..., x n ) s WND, f satsfy condton n Defnton 2.1. A sequence {X, 1} of random varables s sad to be parwse WND f X and X j are WND for all j and also, s sad to be WND f every fnte subset X 1,... X n s WND. In the followng we have example that satsfy the Defnton 2.1. Defnton 2.2. (Lu [9]) The random vector X = (X 1,..., X n ) s sad to be extended negatve dependent (END) f there exsts a constant M 2 > 0 such that both and P(X 1 x 1,..., X n x n ) M 2 P(X 1 > x 1,..., X n > x n ) M 2 P(X x ) P(X > x ) holds for n 1 and x 1,..., x n R. It s obvous that the sequence {X, 1} s negatve dependent (ND) f M 2 = 1. Example 2.3. Let random vector X = (X 1,..., X n ) have n-dmensonal Farle-Gumble-Morgenstern (FGM) dstrbuton as F 1,,n (x 1,, x n ) = F (x ) 1 + α j F (x )F j (x j ), 1jn where F = 1 F, = 1,, n, are correspondng margnal dstrbutons and α j are real numbers chosen such that F 1,,n s a proper n-dmensonal dstrbuton. It s easy to show that the n-varate FGM famly s END wth M 2 = 1 + α j. 1jn Remark 2.4. If X = (X 1,..., X n ) s a WND random vector then s END wth M 2 1. In fact the WND condton presented n Defnton 2.1 s a useful crteron for characterzaton class of END random varables. We lst some basc propertes of WND based on our results. Let X = (X 1,..., X n ) and Y = (Y 1,..., Y n ) be two WND random vectors, then P 1. If h 1 ( ),..., h n ( ) are all strctly monotone real functons then h 1 (X 1 ),..., h n (X n ) are WND. P 2. If h 1 ( ),..., h n ( ) are non-negatve real functon then for all n 2, then there exsts M 1 1 such that E( h (X )) M 1 E(h (X )). In partcular, f h (x ) = e tx, for all 1 and some real t, then E ( n etx ) M1 n E(etX ). P 3. If X Y (X and Y are ndependent ) then random vector (X 1,..., X n, Y 1,..., Y n ) s WND.

3 Nader et al. / Flomat 30:12 (2016), P 4. If X Y and Y s a postve random vector then random vector (X 1 Y 1,..., X n Y n ) s END. P 5. If X Y then (X 1 + Y 1,..., X n + Y n ) s END. P 6. If S n = X, then f Sn (s) M 1 f (n), where f (n) s n-convoluton ndependent of random varables. P 7. For all j, there exsts M 1 1 such that Cov(X, X j ) (M 1 1) F (x)f j (y)dxdy. R R P 8. Let random vectors X and Y have jont denstes f 1 (x 1,..., x n ) and f 2 (x 1,..., x n ) respectvely that havng fxed margnal denstes. Then f α (x 1,..., x n ) = α f 1 (x 1,..., x n ) + (1 α) f 2 (x 1,..., x n ) for 0 α 1, s stll WND. P 9. Any subset of WND random varables s WND. Proof. Here, we prove the property P 10 as the followng. One can easly show the propertes P 1 P 9. Let A and B are parttons of {1, 2, 3,..., n}, n(b) = #B and dx j = dx j1, dx j2,, dx jn(b) correspondng to partton B then, f (x, A) =... f (x, A, x j, j B)dx j M 1 R R = M 1 f (x ). A R... f (x ) f j (x j )dx j R A B Theorem 2.5. Let {X n, n 1} be a absolutely contnuous sequence of WND r-dmensonal random vectors wth sequence of densty functons { f n, n 1} such that there exsts constant M > 0, n : f n M a.e. and f n f a.e., where f s densty functon of random r-dmensonal vector X. Then X s END. Proof. Consder random vector X n = (X 1n, X 2n,..., X rn ) and, f n and f be densty functons random varable X n and X for 1 r respectvely, we can wrte lm f n(x ) = lm... f n (x 1,..., x r )dx 1... dx 1 dx dx r n n R R =... f (x 1,..., x r )dx 1... dx 1 dx dx r = f (x ), (by domnated conver ence theorem). R R Therefore exsts M 1 1 such that f (x 1,..., x r ) = lm n f n (x 1,..., x r ) M 1 r r lm f n(x ) = M 1 f (x ). n So, random vector X s WND, hence by Remark 2.4 X s END. Defnton 2.6. A sequence {X, 1} of random varables s sad to be stochastcally domnated by a random varable X f there exsts a postve constant C such that P( X > x) CP( X > x), for all x 0 and n 1. Lemma 2.7. Let {X, 1} be a sequence of random varables whch s stochastcally domnated by a random varable X. For any β > 0 and b > 0, the followng two statements hold: E X β I( X b) C 1 [E X β I( X b) + b β P( X > b)], E X β I( X > b) C 2 E X β I( X > b), where C 1 and C 2 are postve constants. Consequently, E X β CE X β, where C s a postve constant. Proof. The proof can be found n Lemma of Wu [18]. So the detals are omtted.

4 3. Complete Convergence Theorems Nader et al. / Flomat 30:12 (2016), In ths secton, we derve the complete convergence for weghted sums of WND random varables under some sutable condtons.throughout ths part, the symbol C denotes constant whch s not necessarly the same one n each appearance. The followng theorem concernng the rate of complete convergence of weghted sum n a nx under some sutable condtons on the coeffcents. Theorem 3.1. Let {X, 1} be a sequence of WND random varables whch s stochastcally domnated by a random varable X and {a n, 1 n, n 1} be an array of real numbers wth n a2 n = O(n). If E X α <, 0 < α < 2. When 1 α < 2, assumes that EX = 0, 1. Then for ε > 0, n 1 P max a n X > εn 1/α <. Proof. Let A n = max a n X > εn 1/α and smlarly A n = max a n X > εn1/α n whch X 1, X 2,..., X n are ndependent random varables wth X st = X (means, equal n dstrbuton) for each. Then P (A n ) = f (x 1,..., x n )dx 1... dx n A n M 1 f 1 (x 1 )... f n (x n )dx 1... dx n (by De f nton 2.1) A n = M 1 f 1 (x 1)... fn(x n )dx 1... dx n = M 1 P(A n), where enough to show that n 1 P(A n) <. For any 1, defne A n X (n) = X I( X n1/α ) + n 1/α I(X > n 1/α ) n 1/α I(X < n 1/α ), T (n) k = n 1/α It s easy to show that a n (X (n) EX (n) ). max n a n X > n1/α ε X > n1/α max a n X (n) > n1/α ε. Then for every ε > 0, n 1 P max a n X > n1/α ε + n 1 P( X > n1/α ) n 1 P max = I + II. T (n) k > ε max n 1/α a n EX (n)

5 Nader et al. / Flomat 30:12 (2016), Frst we show that n 1/α max a n EX (n) 0. () When 0 < α < 1, then n 1/α max Ea n X (n) n 1/α a n E X (n) n 1/α a n { E X I( X n1/α ) + n 1/α P( X > n1/α ) } Snce Cn 1/α a n { E X I( X n 1/α ) + n 1/α P( X > n 1/α ) + n 1/α P( X > n 1/α ) } ( by Lemma 2.7 ) = Cn 1 1/α { E X I( X n 1/α ) + 2n 1/α P( X > n 1/α ) } ( by Hölder nequalty a n = O(n) ) Cn 1 1/α E X I((k 1) 1/α < X k 1/α ) + 2CnP( X > n 1/α ). k 1 1/α E X I((k 1) 1/α < X k 1/α ) Therefore, by Kronecker s lemma, we can conclude that n 1 1/α E X I((k 1) 1/α < X k 1/α ) 0 as n, and by Lebesgue domnated convergence theorem, we have kp((k 1) 1/α < X k 1/α ) E X 1/α <. np( X > n 1/α ) = nei( X α > n) E X α I( X α > n) 0 as n. () When 1 α < 2, we have by E X α < that n 1/α max Ea n X (n) n 1/α a n EX (n) n 1/α a n { EX I( X n1/α ) + n 1/α P( X > n1/α ) } n 1/α a n { E X I( X > n1/α ) + n 1/α P( X > n1/α ) } ( by EX = 0 ) Cn 1/α a n { E X I( X > n 1/α ) + n 1/α P( X > n 1/α ) } ( by Lemma 2.7 ) Cn 1 1/α E X I( X > n 1/α ) + np( X > n 1/α ) Cn 1 1/α E X I( X > n 1/α ) + Cn 1 1/α E X I( X > n 1/α ) = 2Cn 1 1/α E X I( X > n 1/α ) = 2Cn 1 1/α E X α X 1 α I( X > n 1/α ) 2CE X α I( X > n 1/α ) 0 as n.

6 Hence, () and () follow that for large enough n and ε > 0 n 1/α max Ea n X (n) < ε/2. Thus, we need only to prove I < and II <. By E X α <, we can get that Nader et al. / Flomat 30:12 (2016), I = n 1 P( X > n1/α ) C P( X > n 1/α ) CE X α <. For 0 < α < 2 and by settng Y = X (n) II ( n 1 P max T (n) k > ε ) 2 n1/α EX (n), we have 4 n 1 2/α Var ε 2 a n Y (by Kolmo orov nequalty) 4 2 n 1 2/α E ε 2 a n Y = 4 ε 2 n 1 2/α n 1 2/α n 1 2/α a 2 n a 2 n ( by Lemma 2.7 ) a 2 n E X (n) 2 { E X 2 I( X n1/α ) + n 2/α P( X > n1/α ) } { E X 2 I( X n 1/α ) + 2n 2/α P( X > n 1/α ) } The proof of Theorem 3.1 has completed. n { 2/α E X 2 I( X n 1/α ) + 2n 2/α P( X > n 1/α ) } n 2/α E X 2 I((k 1) 1/α < X k 1/α ) + 2E X α E X 2 I((k 1) 1/α < X k 1/α ) n 2/α + 2E X α C n=k k 1 2/α E X 2 I((k 1) 1/α < X k 1/α ) + 2E X α ( by n 2/α < n=k x 2 α dx = Ck 1 2/α < ) k 1 kp((k 1) 1/α < X k 1/α ) + 2E X α CE X α <.

7 Nader et al. / Flomat 30:12 (2016), Corollary 3.2. Let {X, 1} be a sequence of WND random varables whch s stochastcally domnated by a random varable X and {a n, 1 n, n 1} be an array of real numbers wth n a2 = O(n). If E X α <, 0 < α < 2. When 1 α < 2, assumes that EX = 0, 1. Therefore n lm a X = 0 a.s. n n 1/α Proof. Let a n = a for n and a n = 0 for > n n Theorem 3.1. Hence, we have for all ε > 0 > n 1 P max 2 a X > εn r+1 1 1/α = n 1 P max r=0 n=2 r a X > εn 1/α 2 r+1 1 (2 r+1 1) 1 P max r=0 n=2 r a X > ε2 (r+1)/α 1 P 2 max 1k2 a X > ε2 r (r+1)/α. It follows from Borel-Cantell Lemma, max 1k2 a X r lm = 0. a.s. r 2 (r+1)/α For all postve ntegers n, there exsts a non-negatve nteger r 0 such that 2 r0 1 n < 2 r 0. Thus n 1/α a X max 2 r 0 1 n<2 r 0 n 1/α a X max 2 (r 0 1)/α a X 2 r 0 1 n<2 r 0 r max 1r<2 r a X α 0 as r 0. Ths completes the proof. 2 (r 0+1)/α r=1 Random weghtng s an emergng computng method n statstcs, t has been put forward by Zheng [19]. Snce ths method has many advantages, t drew much attenton of statstcans. Gao et al. [5] studed law of large numbers for sample mean of random weghtng estmate for..d random varables. In ths secton we determne almost sure convergence for sample mean of random weghtng estmate for WND random varables. Theorem 3.3. The random weghtng estmate of sample mean X n s T n = n β X such that n β = 1 for random varables 0 β 1 a.s., under the condtons of Corollary 3.2, T n X n n 1/α 0 as n, where Θ = β 1 n a.s. Proof. We can wrte T n X = n Θ X and n Θ2 = O(n) a.s. hence condtons Corollary 3.2 hold when Θ = a a.s. for 1 and 0 < α < 2. For that reason, ths completes the proof of Theorem 3.3. Remark 3.4. Theorem 3.3 extends and mproves of the Theorem 2 obtaned by Gao et al. [5].

8 Nader et al. / Flomat 30:12 (2016), Examples In ths secton, some useful examples are dscussed. In each case, Defnton 2.1 s provded. A copula s a functon that lnks unvarate margnal dstrbutons to the full multvarate dstrbuton and t s a general tool for assessng the dependence structure of random varables. For detals see Nelsen [10]. Defnton 4.1. An n-dmensonal copula s a functon C wth doman [0, 1] n f () C(u) = 0 whenever u [0, 1] n has at least one component equal to 0, () C(u) = u whenever u [0, 1] n has all components equal to 1 except the -th one, whch s equal to u, () C s n-ncreasng,.e., for each n-box B = n [u, v ] n [0, 1] n wth u v for each 1,, n, V C (B) = ( 1) N(z) C(z) 0, z n [u,v ] where N(z) = card{k z k = u k }. Some well-known examples of n-copulas are the functons Π, M and W respectvely defned by Π(x 1, x 2,..., x n ) = x 1.x 2....x n M(x 1, x 2,..., x n ) = Mn(x 1, x 2,..., x n ) W(x 1, x 2,..., x n ) = Max(x 1 + x x n n + 1, 0) and whle Π and M are always n-copulas, W s a copula when n = 2. More formally, Sklar s theorem states that any contnuous the jont dstrbuton can be unquely represented by a copula F(x 1, x 2,..., x n ) = C(F 1 (x 1 ), F 2 (x 2 ),..., F n (x n )), where F s the jont dstrbuton to the margnal dstrbuton functons F 1 (x 1 ), F 2 (x 2 ),..., F n (x n ). Recall that the copula densty c s the dervatve of C wth respect to each of ts arguments as: c(u 1, u 2,..., u n ) = n u 1..., u n C(u 1,..., u n ). By usng the chan rule, the densty functon f (x 1, x 2,..., x n ) correspondng to the copula C(x 1, x 2,..., x n ) s, f (x 1, x 2,..., x n ) = c(f 1 (x 1 ), F 2 (x 2 ),..., F n (x n )) f (x ) where f 1 (x 1 ), f 2 (x 2 ),..., f n (x n ) are the margnal denstes of f. Lemma 4.2. Let X = (X 1,..., X n ) be a random vector havng to jont dstrbuton functon F(x 1,..., x n ) and absolute contnuous margnals F 1,..., F n, wth correspondng copula functon C(u 1,, u n ). If c(u 1,..., u n ) M 1 for some M 1 1, then X s WND. Proof. The proof of the lemma s standard, so we omt the detals. Example 4.3. Suppose that C(u 1,..., u n ) = ( n u ) exp{β n (1 u )} for β [ 1, 1]. Then C s a absolutely contnuous copula whch was ntroduced by Čeleboǧlu [2] and also Cuadras [3] has studed some of t s propertes. It s easy to check that f β [ 1, 1] there exsts M 1 = exp( β ) 1 such that Lemma 4.2 holds, for n 2. Therefore, all of the dstrbutons wth gven margns F 1, F 2,..., F n generated by ths copula are WND.

9 Nader et al. / Flomat 30:12 (2016), Example 4.4. (The Farle-Gumble-Morgenstern n-copulas). Let (X 1,..., X n ) be random vector wth the followng jont dstrbuton: F(x 1, x 2,..., x n ) = F (x ) 1 + θ j1 j 2...,j l F j1 (x j1 ) F j2 (x j2 )... F jl (x jk ) ( θ j 1 j 2...j l 1). l=2 1j 1 <...<j l n Where F jk = 1 F jk for l = 2,..., n. Va Sklar s Theorem, t s easy to show that, the copula densty s c(u 1, u 2,..., u n ) = 1 + l=2 1j 1 <...<j l n It s easy to show that c(u 1, u 2,..., u n ) 1 + θ j1 j 2...j l (1 2u j1 )(1 2u j2 )... (1 2u jl ). l=2 1j 1 <...<j l n θ j1 j 2...j l, so there exsts M 1 1 such that Lemma 4.2 holds. Therefore, all of the dstrbutons wth gven margns F 1, F 2,..., F n generated by ths copula are WND. Example 4.5. (The Drchlet dstrbuton) Let (X 1,..., X n ) denotes random vector wth the followng jont densty functon: f(x 1,..., x n ) = Γ( γ ) Γ(γ ) If 0 γ 1 for = 1, 2,..., n and x γ 1, x 0, x = 1. γ 1 then X s WND. Acknowledgments The authors are most grateful to the Edtor and anonymous referees for the careful readng of the manuscrpt and valuable suggestons that helped n sgnfcantly mprovng an earler verson of ths paper. References [1] M. Amn, H. Nl, A. Bozorgna, Complete convergence of movng-average proceeses under negatve dependent Sub-Gussan assumptonns, Bull. Iran. Math. Soc. 38 (2012), [2] S.A.Čeleboǧlo, Way of generatng comprehnsve copulas, J. Ins. Sc. Tech, Gaz Un. 10 (1997), [3] C.M. Cuadras, Constructng copula functons wth weghted geometrc means, J. Statst. Plann. Infer 139(2009), [4] X. Deng, M. Ge, X. Wang, Y. Lu, Y. Zhou, Complete convergence for weghted sums of a class of random varables, Flomat 28:3 (2014), [5] S. Gao, J. Zhang, T. Zhou, Law of large numbers for sample mean of random weghtng estmate, Info. sc. 155 (2003), [6] P.L. Hsu, H. Robbns, Complete convergence and the law of large numbers, Proc. Nat. Acad. 33 (1974), [7] M.H. Ko, On the complete convergence for negatvely assocated random felds, Ta. J. Math. 15(1) (2011), [8] H.Y. Lang, C. Su, Complete convergence for weghted sums of NA sequences, Statst. Probab. Lett. 45 (1999), [9] L. Lu, Precse large devatons for dependent random varables wth heavy tals, Statst. Probab. Lett. 79 (2009), [10] R. Nelsen, An ntroducton to copulas, Sprnger, New York, [11] A. Jun, Y. Deme, Complete convergence of weghted sums for ρ -mxng sequence of random varables, Statst. Probab. Lett. 78(2008), [12] V. Ranjbar, M. Amn, A. Bozorgna, Asympotc behavor of convoluton of dependent random varables wth heavy-tal dstrbuton, Tha. J. Math. 7 (2009), [13] V. Ranjbar, M. Amn, J. Geluk, A. Bozorgna, Asympotc behavor of product of two heavy-tal dependent random varables, Acta. Math. Sn(Englsh Seres) 29(2) (2013), [14] A. Sklar, Fonctons de répartton á n dmen-sons e leurs marges, Publcatons de l Insttut de Statstque de l Unvverste de Pars 8 (1959), [15] S.H. Sung, Complete convergence for weghted sums of negatvely dependent random varables, Stat. Paper. 53 (2012),

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