Hyun-Chull Kim and Tae-Sung Kim

Transcription

1 Commu. Korea Math. Soc ), No. 3, pp A CENTRAL LIMIT THEOREM FOR GENERAL WEIGHTED SUM OF LNQD RANDOM VARIABLES AND ITS APPLICATION Hyu-Chull Kim ad Tae-Sug Kim Abstract. I this paper we derive the cetral limit theorem for aiξi, where {ai, 1 i } is a triagular array of oegative umbers such that sup a2 i <, max 1 i a i 0 as ad ξ is are a liearly egative quadrat depedet sequece. We also apply this result to cosider a cetral limit theorem for a partial sum of a geeralized liear process X = j= a k+jξ j. 1. Itroductio ad results Lehma[6] itroduced a simple ad atural defiitio of positive egative) depedece: A sequece {ξ i, i Z} of radom variables is said to be pairwise positive egative) quadrat depedet pairwise PQDNQD)) if for ay real α i, α j ad i j, P ξ i > α i, ξ j > α j ) ) P ξ i > α i )P ξ j > α j ). A cocept stroger tha PQDNQD) was itroduced by Newma[7]: A sequece {ξ i, i Z} of radom variables is said to be liearly positiveegative) quadrat depedetlpqdlnqd)) if for every pair of disjoit subsets A, B Z ad positive r j s 1.1) r j ξ j are P QD NQD). i A r i ξ i ad j B Newma[7] established the cetral limit theorem for a strictly statioary LPQDor LNQD) process ad Birkel[2] also obtaied a fuctioal cetral limit theorem for LPQD process which is used to obtai the fuctioal cetral limit theorem for LNQD process. Kim ad Received March 31, Mathematics Subject Classificatio: 60F05, 60G10. Key words ad phrases: cetral limit theorem, liear process, liearly egative quadrat depedet, uiformly itegrable, triagular array. This work was supported by KOSEF GratR ).

2 532 Hyu-Chull Kim ad Tae-Sug Kim Baek[5] exteded this result to a statioary liear process of the form X k = j=0 a jξ k j, where {a j } is a sequece of real umbers with j=0 a j < ad {ξ k } is a strictly statioary LPQD process with Eξ i = 0, 0 < Eξi 2 <, which ca be exteded to the LNQD case by similar method. I this paper, we derive a cetral limit theorem for a liearly egative quadrat depedet sequece i a double array, weakeig the strictly statioarity assumptio with uiform itegrability see Theorem 1.1 below) ad apply this result to obtai a cetral limit theorem for a partial sum of a liear process X = j= a k+jξ j geerated by liearly egative quadrat depedet sequece {ξ j } see Theorem 1.2 below). Theorem 1.1. Let {a i, 1 i } be a triagular array of oegative umbers such that 1.2) sup a 2 i < ad 1.3) max 1 i a i 0 as. Let {ξ i } be a cetered sequece of liearly egative quadrat depedet radom variables such that 1.4) {ξ 2 i } is a uiformly itegrable family, 1.5) Var ad 1.6) j: i j u a i ξ i ) = 1, Covξ i, ξ j ) 0 as u uiformly i i 1 see Cox ad Grimmet[3]). The D a i ξ i N0, 1) as. Remark. Theorem 1.1 exteds the Newma s[7] cetral limit theorem for strictly statioary LNQD sequece from equal weights to geeral weights, weakeig at the same time the assumptio of statioarity.

3 A cetral limit theorem for geeral weighted sum 533 Corollary 1.1. Let {ξ i } be a cetered sequece of liearly egative quadrat depedet radom variables such that {ξi 2 } is a uiformly itegrable family ad let {a i, 1 i } be a triagular array of oegative umbers such that a 2 i 1.7) sup σ 2 <, a i 1.8) max 0 as, 1 i σ where σ 2 = Var a iξ i ). If 1.6) holds the, as 1 D 1.9) a i ξ i N0, 1). σ Theorem 1.2. Let {a j, j Z} be a sequece of oegative umbers such that j a j < ad let {ξ j, j Z} be a sequece of liearly egative quadrat depedet radom variables which is uiformly itegrable i L 2 ad satisfies 1.6). Let X k = a k+j ξ j ad S = X i. Assume j= 1.10) if 1 1 σ 2 > 0, where σ 2 = VarS ). The 1.11) S σ D N0, 1) as. This result is a extesio of Theorem i Ibragimov ad Liik[4] from i.i.d. to the liearly egative quadrat depedece case by addig the coditio 1.6) ad improves o Kim ad Baek s[5] result about cetral limit theorem for aliear processes geerated by LNQD sequeces. 2. Proofs We starts with the followig lemma.

4 534 Hyu-Chull Kim ad Tae-Sug Kim Lemma 2.1. Newma[8]) Let {Z i, 1 i } be a sequece of liearly egative quadrat depedet radom variables with fiite secod momets. The E expit Z j ) E expitz j ) Ct 2 Var Z j ) VarZ j ) for all t R, where C > 0 is a arbitrary costat, ot depedig o. Proof of Theorem 1.1. Without loss of geerality we assume that a i = 0 for all i > ad supeξ 2 = M <. For every 1 a < b ad 1 u b a we have, after a simple maipulatios, 2.1) 0 b b u a i i=a j=i+u sup k j: k j u a j Covξ i, ξ j ) b Covξ k, ξ j ) i=a I particular by defiitio of LNQD, we also have b ) b Var a i ξ i M a 2 i. i=a We shall costruct ow a triagular array of radom variables {Z i, 1 i } for which we shall make use of Lemma 2.1. Fix a small positive ɛ ad fid a positive iteger u = u ɛ such that, for every u + 1 u 0 ɛ. b a i j=i+u i=a a j Covξ i, ξ j ) ) This is possible because of 2.1) ad 1.6). Deote by [x] the iteger part of x ad defie Y j = uj+1) i=uj+1 K = [ 1 ɛ ] a i ξ i, j = 0, 1,, A j = i : 2Kj i < 2Kj + K, CovY i, Y,i+1 ) 2 K a 2 i ). 2Kj+K i=2kj VarY i ).

5 A cetral limit theorem for geeral weighted sum 535 Sice 2CovY i, Y,i+1 ) VarY i ) + VarY,i+1 ), we get that for every j the set A j is ot empty. Now we defie the itegers m 1, m 2,, m recurretly by m 0 = 0: m j+1 = mim{m; m > m j, m A j } ad defie Z j = m j+1 i=m j +1 Y i, j = 0, 1,, j = {um j + 1) + 1,, um j+1 + 1)}. We observe that Z j = k j a k ξ k, j = 0, 1,. By defiitio of LNQD Z j s are liearly egative quadrat depedet, ad from the fact that m j 2Kj 1) ad m j+1 K2j + 1) every set j cotais o more tha 3 Ku elemets ad m j+1 /m j 1 as j. Hece, for every fixed positive ɛ by 1.2) 1.5) the array {Z j : i = 1,, ; 1} satisfies the Lideberg s coditiosee Stout [9]). It remais to observe that by Lemma 2.1 ad costructio. E expit ct 2 {Var ct 2 {2 u ct 2 {4 Z j ) Z j ) ct 2 {4ɛ + 8 K = ct 2 {4ɛ + 8 K E expitz j ) VarZ j ) } 2 CovZ i, Z,i+1 ) ) + 2 a i j=i+u a j Covξ i, ξ j ) + 2 VarY i )} uj+1) Var a i ξ i )} i=uj+1 j=i+2 CovZ i, Z j ) )} CovY,mj, Y,mj +1) }

6 536 Hyu-Chull Kim ad Tae-Sug Kim ct 2 {4ɛ + 8M K c 1 t 2 ɛ{1 + sup c 2 t 2 ɛ. uj+1) i=uj+1 a 2 i} a 2 i } Now the proof is complete by Theorem 4.2 i Billigslely[1]. Proof of Corollary 1.1. Let A i = a i σ. The we have max A i 0 as, 1 i sup Var A 2 i <, A i ξ i ) = 1. Hece, by Theorem 1.1 the desired result 1.11) follows. Proof of Theorem 1.2. First ote that j a2 j < ad without restrictig the geerality, we ca assume sup Eξk 2 = 1. Let S = X k = k=1 j= k=1 a k+j )ξ j. I order to apply Theorem 1.1, we fix W such that j >W a 2 j < 3 ad take k = W +. The S σ = j k k=1 = T + U say). a k+j ) ξ j σ + j >k k=1 a k+j ) ξ j σ By Cauchy Schwarz iequality ad assumptios we have the followig estimate VarU ) ) ξ j a k+j σ j >k Var k=1

7 A cetral limit theorem for geeral weighted sum 537 which yields j >k k=1 σ 2 2 σ 2 2 σ 2 a k+j /σ ) 2 Eξj 2 j >k k=1 j >k j >W a 2 j a 2 k+j ) a 2 j 1 σ 2 0 as, 2.2) U 0 i probability as. It remais oly to prove that T Billigsley[1]. Put D N0, 1) by Theorem 4.1 of 2.3) a k = a k+j σ. From assumptio j a j < a j > 0), 1.10) ad 2.3) we obtai sup <k< a k+j σ 0 as, max a k 0 as, 1 k Hece, by Theorem 1.1 sup a 2 k <. k=1 2.4) T D N0, 1) ad from 2.2) ad 2.4) the desired result 1.10) follows. Ackowledgemets. The authors wish to thak the referee for a very thorough review of this paper ad Tae-Sug Kim is the correspodig author.

8 538 Hyu-Chull Kim ad Tae-Sug Kim Refereces [1] P. Billigsley, Covergece of Probability Measures, Wiley, New York, [2] P. Birkel, A fuctioal cetral limit theorem for positively depedet radom variables, J. Multivariate Aal ), [3] J. T. Cox ad G. Grimmett, Cetral limit theorems for associated radom variables ad the percolatio model, A. Probab ), [4] I. A. Ibragimov ad Yu. V. Liik, Idepedet ad Statioary Sequeces of Radom Variables, Volters, Groige, [5] T. S. Kim ad J. L. Baek, A cetral limit theorem for statioary liear processes geerated by liearly positive quadrat depedet process, Statist. Probab. Lett ), [6] E. L. Lehma, Some cocepts of depedece, A. Statist ), [7] C. M. Newma, Asymptotic idepedece ad limit theorems for positively ad egatively depedet radom variables, I: Tog, Y. L.Ed.), Stochastics ad Probability ), Ist. Math. Statist. Hayward, C.A.) [8], Normal fluctuatios ad the FKG iequalities, Comm. Math. Phys ), [9] W. F. Stout, Almost Sure Covergece, Academic Press, New York, Hyu-Chull Kim Divisio of Computer ad Iformatio Sciece Daebul Uiversity Yeogam , Korea kimhc@mail.daebul.ac.kr Tae-Sug Kim Divisio of Mathematics ad Iformatioal Statistics ad Istitute of Basic Natural Sciece WoKwag Uiversity Iksa , Korea starkim@wokwag.ac.kr