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Let $X_n$ be independent random elements in the Skorohod space $D([0,1];E)$of c\`{a}dl\`{a}g functions taking values in a separable Banach space $E$. Let$S_n=\sum_{j=1}^nX_j$. We show that if $S_n$ converges in finite dimensionaldistributions to a c\`{a}dl\`{a}g process, then $S_n+y_n$ converges a.s.pathwise uniformly over $[0,1]$, for some $y_n\in D([0,1];E)$. This resultextends the It\^{o}-Nisio theorem to the space $D([0,1];E)$, which issurprisingly lacking in the literature even for $E=R$. The main difficulties ofdealing with $D([0,1];E)$ in this context are its nonseparability under theuniform norm and the discontinuity of addition under Skorohod's $J_1$-topology.We use this result to prove the uniform convergence of various seriesrepresentations of c\`{a}dl\`{a}g infinitely divisible processes. As aconsequence, we obtain explicit representations of the jump process, and ofrelated path functionals, in a general non-Markovian setting. Finally, weillustrate our results on an example of stable processes. To this aim we obtainnew criteria for such processes to have c\`{a}dl\`{a}g modifications, which mayalso be of independent interest.

On the uniform convergence of random series in Skorohod space and representations of c\`{a}dl\`{a}g infinitely divisible processes