A CENTRAL LIMIT THEOREM FOR LINEAR PROCESSES UNDER LINEAR NEGATIVELY QUADRANT DEPENDENCE

Abstract

Abstract. In this paper we establish a central limit theorem forweighted sums of Y n =P ni =1 a n;i X i , where fa n;i ;n 2 N; 1 • i • ng is an array of nonnegative numbers such that sup n‚ 1 P ni =1 a 2 n;i <1 , max 1 •i•n a n;i ! 0 and fX i ;i 2 Ng is a sequence of linearnegatively quadrant dependent random variables with EX i = 0and EX 2 i < 1 . Using this result we will obtain a central limittheorem for partial sums of linear processes. 1. IntroductionFor a sequence fa n ;n ‚ 1 g of real numbers the limit superior is de-flnedbyinf r‚ 1 sup n‚ a n andisdenotedbylimsup n!1 a n . Let fX n ;n ‚ 1 g be a sequence of random variables and fa n;k ;n ‚ 1 ; 1 • k • ng bean array of real numbers. The weighted sumsP nk =1 a n;k X k can play animportant role in various applied and theoretical problems, such as thoseof the least squares estimators(see Ka°es and Bhaskara Rao(1982)) andM-estimates(see Rao and Zhao(1992)) in linear models, the nonpara-metric regression estimators(see Priestley and Chao(1972)), etc. So thestudy of the central limit theorem is every important and signiflcant.Two random variables