A central limit theorem for moving average process with negatively associated innovation

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1 Iteratioal Mathematical Forum, 1, 2006, o. 10, A cetral limit theorem for movig average process with egatively associated iovatio MI-HWA KO Statistical Research Ceter for Complex Systems Seoul Natioal Uiversity, Seoul , Korea kmh@srccs.su.ac.kr DAE-HEE RYU Departmet of Computer Sciece ChugWoo Uiversity, Chugam, , Korea, rdh@cwet.ac.kr KWANG-HEE HAN Departmet of Computer Sciece Howo Uiversity, Jeobuk, , Korea, khha@suy.howo.ac.kr Abstract I this paper we derive the cetral limit theorem for a i ξ i, where {a i, 1 i } is a triagular array of oegative umbers such that sup a 2 i <, max 1 i a i 0 as ad the iovatio {ξ i } is a cetered sequece of egatively associated radom variables. We also apply this result to obtai the asymptotic behavior of a partial sum of a movig average process X = j= a k+j ξ j where {a j } is a sequece of oegative umbers such that j= a j <. Mathematics Subject Classificatios : 60F05 ; 60G10 Keywords : Cetral limit theorem; Liear process; Movig average processes; Negative associated; Uiformly itegrable; Triagular array. 1. Itroductio The sequece {ξ i,i N} is called egatively associated(na) if for every pair of disjoit subsets A, B of N ad ay pair of coordiatewise icreasig fuctios f(ξ i,i A),g(ξ j,j B) with Ef 2 (ξ i,i A) < ad Eg 2 (ξ j,j B) <, it holds that Cov(f(ξ i,i A),g(ξ j,j B)) 0.

2 496 M.H. Ko, D.H. Ryu ad K.H. Ha The cocept of NA was itroduced by Joag-Dev ad Proscha(1983). As poited out ad proved by Joag-Dev ad Proscha(1983), a umber of wellkow multivariate distributios possess the NA property, such as multiomial distributio, multivariate hypergeometric distributio, egatively correlated ormal distributio ad joit distributio of raks. Because of their wide applicatios i multivariate statistical aalysis ad reliability theory, the cocepts of egatively associated radom variables have received extesive attetio recetly. We refer to Joag-Dev ad the Proscha(1983) for fudametal properties ad we refer to Newma(1984) for a momet iequality ad the covergece i distributio. Let {ξ i } be a cetered sequece of radom variables ad let {a i, 1 i } be a triagular array of umbers. May statistical procedures produce estimators of the type S = a i ξ i. (1.1) We will derive the covergece i distributio of the type (1.1), where ξ is are NA sequece. Defie a movig average process by X = a k+j ξ j. (1.2) j= where the iovatio {ξ k } is a cetered sequece of radom variables ad {a k } is a sequece of real umbers. I time-series aalysis, this process is of great importace. May importat time-series models, such as the causal ARMA process(brockwell ad Davis, 1987, p. 89), have type (1.2). We shall see that the cetral limit theorem of NA variables of the form (1.2), where ξ js are NA radom variables, ca be obtaied by the study of the weghted sum of NA radom variables of the form (1.1). Our paper orgaized i the followig way : Sectio 2 cotais the mai results ad the statemet of some kow results which will be used i the proof ad Sectio 3 cotais the proof of the mai results. 2. Results Theorem 2.1 Let {a i, 1 i } be a triagular array of oegative umbers such that sup a 2 i <, (2.1)

3 A cetral limit theorem for movig average process 497 ad max 1 i a i 0 as. (2.2) Let {ξ i } be a cetered sequece of egatively associated radom variables such that ad {ξi 2 } is a uiformly itegrable family (2.3) Var( a i ξ i )=1. (2.4) Assume Cov(ξ i,ξ j ) 0 as u uiformly i i 1. (2.5) j: i j u The a i ξ i D N(0, 1) as. (see Cox ad Grimmet(1984)) Remark Theorem 2.1 exteds the Newma s(1984) cetral limit theorem for strictly statioary egatively associated sequece from equal weights to geeral weights, weakeig at the same time the assumptio of statioarity. Corollary 2.1 Let {ξ i } be a egatively associated sequece of cetered 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 sup <, (2.6) σ 2 a i max 0 as. (2.7) 1 i where σ 2 = Var( a i ξ i ). If (2.5) holds, the, as 1 a i ξ i D N(0, 1). (2.8)

4 498 M.H. Ko, D.H. Ryu ad K.H. Ha Theorem 2.2 Let {a j,j Z} be a sequece of oegative umbers such that j a j < ad let {ξ j,j Z} be a cetered sequece of egatively associated radom variables which is uiformly itegrable i L 2 ad satisfies (2.5). Let X k = a k+j ξ j ad S = X i. Assume j= where σ 2 = Var(S ). The S if 1 1 σ 2 > 0 (2.9) D N(0, 1) as. (2.10) This result is a extesio of Theorem i Ibragimov ad Liik(1971) from the i.i.d. to the egative associatio by addig the coditio (2.5). 3. Proofs We starts with the followig lemma. Lemma 3.1 (Newma, 1980) Let {Z i, 1 i } be a sequece of egatively associated radom variables with fiite secod momets. The E exp(it Z j ) E exp(itz j ) Ct 2 Var( Z j ) Var(Z j ) j=1 j=1 j=1 j=1 for all t R, where C>0is a arbitrary costat, ot depedig o. Proof of Theorem 2.1 Without loss of geerality we assume that a i =0 for all i>ad supeξ 2 = 1. For every 1 a<b ad 1 u b a we have, after a simple maipulatios, 0 sup k b u a i b i=a j=i+u ( j: k j u a j Cov(ξ i,ξ j ) Cov(ξ k,ξ j ) )( b i=a a 2 i ). (3.1) I particular, by (2.5) ad (3.1) there exists a positive costat M, for every 1 a b, b b Var( a i ξ i ) M a 2 i. (3.2) i=a i=a

5 A cetral limit theorem for movig average process 499 We shall costruct ow a triagular array of radom variables {Z i, 1 i } for which we shall make use of Lemma 3.1. Fix a small positive ɛ ad fid a positive iteger u = u ɛ such that, for every u +1 u 0 ( b a i j=i+u a j Cov(ξ i,ξ j ) ) ɛ. (3.3) This is possible because of (3.1) ad (2.5). Deote by [x] the iteger part of x ad defie K =[ 1 ɛ ], Y j = u(j+1) i=uj+1 a i ξ i, j =0, 1,, A j = {i :2Kj i<2kj + K, Cov(Y i,y,i+1 ) 2 K 2Kj+K i=2kj Var(Y i )}. Sice 2Cov(Y i,y,i+1 ) Var(Y i )+Var(Y,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. Let m 0 = O ad ad defie m j+1 = mim{m; m>m j,m A j } Z j = m j+1 i=m j +1 Y i, j =0, 1,, Δ j = {u(m j +1)+1,,u(m j+1 +1)}. We observe that Z j = k Δ j a k ξ k, j =0, 1,, By defiitio of NA Z js are egatively associated. From the fact that m j 2K(j 1) ad m j+1 K(2j + 1) every set Δ j cotais o more tha 3 Ku elemets ad m j+1 /m j 1 as j. Hece, for every fixed positive ɛ by (2.1) - (2.4) the array {Z j : i =1,,; 1} satisfies the Lideberg s coditio(see Petrov(1975), Theorem 22, p.100). It remais to observe that by Lemma 3.1 ad costructio,

6 500 M.H. Ko, D.H. Ryu ad K.H. Ha E exp(it Z j ) E exp(itz j ) j=1 j=1 ct 2 {Var( Z j ) Var(Z j ) } j=1 j=1 2 ct 2 {2( Cov(Z i,z,i+1 ) )+2( Cov(Z i,z j ) )} j=i+2 u ct 2 {4 a i a j Cov(ξ i,ξ j ) +2 Cov(Y,mj,Y,mj +1) } j=i+u j=1 ct 2 {4ɛ + 8 Var(Y i )} K = ct 2 {4ɛ + 8 u(j+1) Var( a i ξ i )} K j=1 i=uj+1 ct 2 {4ɛ + 8M K c 1 t 2 ɛ{1 + sup c 2 t 2 ɛ. u(j+1) j=1 i=uj+1 a 2 i } a 2 i } by (3.2) Now the proof is complete by Theorem 4.2 i Billigslely(1968). Proof of Corollary 2.1 : Let A i = a i. The we have max A i 0 as, 1 i sup A 2 i <, Var( A i ξ i )=1. Hece, by Theorem 2.1 the desired result (2.10) follows. Proof of Theorem 2.2 : First ote that j a 2 j < ad without restrictig the geerality, we ca assume sup Eξk 2 = 1. Let S = X k = ( a k+j )ξ j. k=1 j= k=1

7 A cetral limit theorem for movig average process 501 I order to apply Theorem 2.1, we fix W such that j >W a 2 j < 3 ad take k = W +. The S = ( j k k=1 = T + U (say). a k+j ) ξ j + ( j >k k=1 a k+j ) ξ j ad by Cauchy Schwarz iequality ad assumptios we have the followig estimate Var(U ) ( ) ξ j Var a k+j j >k σ k=1 ( a k+j / ) 2 Eξj 2 j >k k=1 σ 2 ( a 2 k+j ) j >k k=1 2 σ 2 a 2 j j >k 2 σ 2 a 2 j j >W which yields 1 σ 2 0 as U 0 i probability as. (3.4) By Theorem 4.1 of Billigsley(1968) it remais oly to prove that T N(0, 1). Put D a k = j=1 a k+j. (3.5) From assumptio j a j < (a j > 0), (2.9) ad (3.5) we obtai sup <k< j=1 a k+j 0 as, max a k 0 as, 1 k sup a 2 k <. k=1

8 502 M.H. Ko, D.H. Ryu ad K.H. Ha Hece, by Theorem 2.1 T D N(0, 1) (3.6) ad from (3.4) ad (3.6) the desired result (2.10) follows. ACKNOWLEDGEMENTS. The authors would like to thak the referee for careful readig of the mauscript.this work was supported by the SRC/ERC program of MOST/KOSEF (R ) ad the Howo Uiversity research grat i This paper was partially supported by Chug Woo Uiversity i Refereces [1] Billigsley, P.(1968) Covergece of Probability Measures. Wiley, New York. [2] Brockwell, P.J. ad Davis, R.A.(1987) Time Series: Theory ad Methods. Spriger, New York. [3] Cox, J. T. ad Grimmett, G.(1984) Cetral limit theorems for associated radom variables ad the percolatio model. A. Probab [4] Ibragimov, I. A. ad Liik, Yu. V.(1971) Idepedet ad Statioary Sequeces of Radom Variables. Volters, Groige. [5] Joag-Dev, K. Proscha, F. (1983). Negative associatio of radom variables with applicatio. A. Statist [6] Newma, C. M.(1980) Normal fluctuatios ad the FKG iequalities. Comm. Math. Phys [7] Newma, C. M.(1984) Asymptotic idepedece ad limit theorems for positively ad egatively depedet radom variables. I Iequalities i Statistics ad Probability(Y. L. Tog, Ed.), IMS Lecture Notes- Moogragh Series, 5 pp (Istitute of Mathematical Statistics, Hayward, Califoria.) [8] Petrov, V. V.(1975) Sums of idepedet radom variables. Berli, Heidelberg. New York. Received: April 20, 2005

Hyun-Chull Kim and Tae-Sung Kim

Hyun-Chull Kim and Tae-Sung Kim Commu. Korea Math. Soc. 20 2005), No. 3, pp. 531 538 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