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

TL;DR

This paper extends the Itô-Nisio theorem to the Skorohod space of càdlàg functions in Banach spaces, establishing uniform convergence of series representations of infinitely divisible processes and providing new criteria for stable processes to have càdlàg modifications.

Contribution

It introduces a novel uniform convergence result in Skorohod space, enabling explicit series representations of càdlàg infinitely divisible processes, including stable processes.

Findings

Extended the Itô-Nisio theorem to D([0,1];E)

Proved uniform convergence of series representations of càdlàg processes

Derived new criteria for stable processes to have càdlàg modifications

Abstract

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 dimensional distributions 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 result extends the It\^{o}-Nisio theorem to the space $D([0,1];E)$, which is surprisingly lacking in the literature even for $E=R$. The main difficulties of dealing with $D([0,1];E)$ in this context are its nonseparability under the uniform norm and the discontinuity of addition under Skorohod's $J_1$-topology. We use this result to prove the uniform convergence of various series representations of c\`{a}dl\`{a}g infinitely divisible processes. As a consequence, we obtain explicit representations of the jump process, and of related path functionals, in a general non-Markovian setting. Finally, we illustrate our results on an example of stable processes. To this aim we obtain new criteria for such processes to have c\`{a}dl\`{a}g modifications, which may also be of independent interest.