Weak Laws of Large Numbers in Normed Linear Spaces

Abstract

In this paper weak laws of large numbers are proved for random elements (function-valued random variables) in separable normed linear spaces. One result states that for identically distributed random elements $\{V_n\}$ such that the Pettis integral $EV_1$ exists and $E\|V_1\| < \infty$ $\|n^{-1} \sum^n_{k=1} V_k - EV_1\| \rightarrow 0 \text{in probability}$ if and only if $|n^{-1} \sum^n_{k=1} f(V_k) - Ef(V_1)| \rightarrow 0 \text{in probability}$ for each continuous linear functional $f$. The condition of identically distributed random elements $\{V_n\}$ can not be relaxed by just assuming a bound on the moments of $\{\|V_n\|\}$, but a weak law of large numbers is obtained for random elements which need not be identically distributed. Both of these weak laws can also be obtained by assuming only that the space has a Schauder basis such that the weak law of large numbers holds in each coordinate. An application of these results yields a uniform weak law of large numbers for separable Wiener processes on [0, 1].