On the uniform convergence of random series in Skorohod space and representations of cadlag infinitely divisible processes

Details On the uniform convergence of random series in Skorohod space and representations of cadlag infinitely divisible processes On the uniform convergence of random series in Skorohod space and representations of cadlag infinitely divisible processes

2011-11-07

arxiv.org/abs/1111.1682v1

Mathematics

Probability

Scientific paper

Let $X_n$ be independent random elements in the Skorohod space $D([0,1]; E)$ of cadlag functions taking values in a separable Banach space $E$. Let $S_n = \sum_{j=1}^{n} X_j$. We show that if $S_n$ converges in finite dimensional distributions to a cadlag 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 Ito-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 non-separability under the uniform norm and the discontinuity of the addition under Skorohod's $J_1$-topology. We use this result to prove the uniform convergence of various series representations of cadlag 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 cadlag modifications, which may also be of independent interest.

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