Inequalities for $B$-Valued Random Vectors with Applications to the Strong Law of Large Numbers

Abstract

Analogues of the Marcinkiewicz-Zygmund and Rosenthal inequalities for Banach space valued random vectors are proved. As an application some results on the strong law of large numbers are obtained. It is proved that the Marcinkiewicz SLLN holds for every $p$-integrable, mean zero $B$-valued $\mathrm{rv}$ if and only if $B$ is of Rademacher type $p(1 \leq p < 2)$.