A NOTE ON CONVERGENCE OF WEIGHTED SUMS OF RANDOM VARIABLES

ABSTRACT. Under uniform integrability condition, some Weak Laws of large numbers are established for weighted sums of random variables generalizing results of Rohatgi, Pruitt and Khintchine. Some Strong Laws of Large Numbers are proved for weighted sums of pairwise independent random variables generalizing results of Jamison, Orey and Pruitt and Etemadi. KEY WORDS AND PHRASES. Independent and pairwise independent random variables, weighted sums of random variables, uniform integrability, convergence in probability, in mean, strong convergence, random elements in separable Banach spaces. 1980 AMS SUBJECT CLASSIFICATION CODE. 60F05, 60F15. Let Xn, n be a sequence of real random variables defined on a probability space (, B, P) and ank, n _> I, k _> a double array of real numbers. Limit theorems have been studied in the literature for the sequence Z ankXk, n _> of weighted sums k>l of the sequence Xn, n _> under some conditions on the double array of numbers and on the distribution of the sequence Xn, n I. Jamison, Orey and Pruitt [4] studie