Stability for randomly weighted sums of random elements

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.

Keywords

• Randomly weighted sums
• arrays of random elements
• almost sure convergence
• Banach spaces of type p
• law of large numbers.