International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 3, Pages 559-568
doi:10.1155/S0161171299225598

Stability for randomly weighted sums of random elements

Tien-Chung Hu1 and Hen-Chao Chang2

1Department of Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan
 2Institute of Mathematics, Academic Sinica, Taipei 11529, Taiwan

Abstract

Let {Xn:n=1,2,3,…} be a sequence of i.i.d. random elements taking values in a separable Banach space of type p and let {An,i:i=1,2,3,…;n=1,2,3,…} be an array of random variables. In this paper, under various assumptions of {An,i}, the necessary and sufficient conditions for ∑i=1∞An,iXi→0 a.s. are obtained. Also, the necessity of the assumptions of {An,i} is discussed.