Convergence of sums of mixing triangular arrays of random vectors with stationary rows

Abstract

Several authors have studied the weak convergence of the laws of suns of randon variables with the hypothesis of independence replaced by less restrictive properties which are expressed through certain dependence coefficients (see, for example, Ibraginov and Linnik [12], Billingsley [6],[7], Iosifescu and Theodorescu [13], Philipp [15]). In this paper we consider certain mixing conditions (the so-called φ and ψ-mixing) for triangular arrays of random vectors which take values in a separable Banach space and whose rows form stationary finite sequences (see Section 1 for the definitions). Our aim is to give necessary and sufficient conditions for the convergence of the laws of the row sums of such triangular arrays expressed in terms of the individual random vectors and, in principle, without moment assumptions.