An extension of Feller’s strong law of large numbers

Abstract

This paper presents a general result that allows for establishing a link between the Kolmogorov–Marcinkiewicz– Zygmund strong law of large numbers and Feller’s strong law of large numbers in a Banach space setting. Let {X,Xn;n≥1} be a sequence of independent and identically distributed Banach space valued random variables and set Sn=∑i=1nXi,n≥1. Let {an;n≥1} and {bn;n≥1} be increasing sequences of positive real numbers such that limn→∞an=∞ and bn∕an;n≥1 is a nondecreasing sequence. We show that Sn−nEXI{‖X‖≤bn}bn→0almost surelyfor every Banach space valued random variable X with ∑n=1∞P(‖X‖>bn)<∞ if Sn∕an→0 almost surely for every symmetric Banach space valued random variable X with ∑n=1∞P(‖X‖>an)<∞. To establish this result, we invoke two tools (obtained recently by Li, Liang, and Rosalsky): a symmetrization procedure for the strong law of large numbers and a probability inequality for sums of independent Banach space valued random variables.