On the almost sure convergence of weighted sums of random elements in D[0,1] - R. L. Taylor; C. A. Calhoun (International Journal of Mathematics and Mathematical Sciences)
Let {wn} be a sequence of positive constants and Wn=w1+…+wn where Wn→∞ and wn/Wn→∞. Let Wn be a sequence of independent random elements in D[0,1]. The almost sure convergence of Wn−1∑k=1nwkXk is established under certain integral conditions and growth conditions on the weights {wn}. The results are shown to be substantially stronger than the weighted sums convergence results of Taylor and Daffer (1980) and the strong laws of large numbers of Ranga Rao (1963) and Daffer and Taylor (1979).