Stability for randomly weighted sums of random elements (Q1587815)

Summary: Let \(\{X_n: n= 1,2,3,\dots\}\) be a sequence of i.i.d. random elements taking values in a separable Banach space of type \(p\) and let \(\{A_{n,i}: i= 1,2,3,\dots\); \(n= 1,2,3,\dots\}\) be an array of random variables. Under various assumptions on \(\{A_{n,i}\}\), necessary and sufficient conditions for \(\sum^\infty_{i=1} A_{n,i} X_i\to 0\) a.s. are obtained. Also, the necessity of the assumptions on \(\{A_{n,i}\}\) is discussed.