Convergence of probability measures on separable Banach spaces

Abstract:

The following result follows immediately from a general theorem on the convergence of probability measures on separable Banach spaces: On the space $C[0,1]$ there exists a norm $p(x)$ equivalent to the ordinary norm such that if ${\xi _1}(t), \ldots ,{\xi _n}(t), \ldots$ and $\xi (t)$ are continuous random processes $(0 \leqslant t \leqslant 1)$ and for any finite set of points ${t_1}, \ldots ,{t_k} \subset [0,1]$ the joint distribution of $p({\xi _n}),{\xi _n}({t_1}), \ldots ,{\xi _n}({t_k})$ converges to the joint distribution of $p(\xi ),\xi ({t_1}), \ldots ,\xi ({t_k})$ then ${\xi _n}(t)$ converges weakly to $\xi (t)$.