Studia Scientiarum Mathematicarum Hungarica 43 (1), 33–46 (2006) DOI: 10.1556/SScMath.43.2006.1.2 ON THE ASYMPTOTIC INDEPENDENT REPRESENTATIONS FOR SUMS OF SOME WEAKLY DEPENDENT RANDOM VARIABLES RAFIK AGUECH 1 , SANA LOUHICHI 2 and SOFYEN LOUHICHI 2 1 Facult´ e de Sciences de Monastir, D´ epartement de Math´ ematiques, BD de L’Environnement 5000 Monastir, Tunisie e-mail: rafikaguech@ipeit.rnu.tn 2 Universit´ e de Paris-Sud, Probabilit´ es, Statistique et Mod´ elisation, Bˆat. 425, 91405 Orsay Cedex, France e-mail: sana.louhichi@math.u-psud.fr Communicated by E. Cs´aki (Received December 18, 2003; accepted April 4, 2005) Abstract Let, for each n ∈ ,(X i,n ) 0 i n be a triangular array of stationary, centered, square integrable and associated real valued random variables satisfying the weakly dependence condition lim N→N 0 lim sup n→+∞ n n r=N Cov (X0,n,Xr,n) = 0; where N 0 is either infinite or the first positive integer N for which the limit of the sum n n r=N Cov (X0,n,Xr,n) vanishes as n goes to infinity. The purpose of this paper is to build, from (X i,n ) 0 i n , a sequence of independent random variables ( ˜ X i,n ) 0 i n such that the two sums n i=1 X i,n and n i=1 ˜ X i,n have the same asymptotic limiting behavior (in distribution). 1. Introduction Let, for n ∈ N,(X i,n ) 0 i n be a triangular array of row-wise station- ary, centered and square integrable real valued random variables. Put S n = ∑ n i=1 X i,n . Our main task in this paper is to find a sequence ( ˜ X i,n ) of 2000 Mathematics Subject Classification. Primary 60F05. Key words and phrases. Central limit theorem, independent representation, weakly dependence, associated random variables. 0081–6906/$ 20 c  2006 Akad´ emiai Kiad´o, Budapest