Laws of Large Numbers for Tight Random Elements in Normed Linear Spaces

Abstract

A strong law of large numbers is proved for tight, independent random elements (in a separable normed linear space) which have uniformly bounded $p$th moments $(p > 1)$. In addition, a weak law of large numbers is obtained for tight random elements with uniformly bounded $p$th moments $(p > 1)$ where convergence in probability for the separable normed linear space holds if and only if convergence in probability for the weak linear topology holds.