Strong laws of large numbers for weighted sums of random elements in normed linear spaces (Q1822822)

Let \(\{V_ n,n\geq 1\}\) be a sequence of independent random elements in a real separable normed linear space and let \(\{a_ n\}\), \(\{b_ n\}\) be sequences of constants with \(0<b_ n\uparrow \infty\). The main results discussed here provide conditions under which \(\{a_ n(V_ n-E V_ n)\), \(n>1\}\) obeys a strong law of large numbers of the form \[ \sum^{n}_{j=1}a_ j(V_ j-E V_ j)/b_ n\to 0\quad a.s. \] It is further shown that Feller's generalization of the Marcinkiewicz-Zygmund SLLN holds for random elements in a real separable Rademacher type p \((1<p\leq 2)\) Banach space.