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.
"Laws of Large Numbers for Tight Random Elements in Normed Linear Spaces." Ann. Probab. 7 (1) 150 - 155, February, 1979. https://doi.org/10.1214/aop/1176995156