ON THE UNIFORM CONVERGENCE OF RANDOM SERIES IN SKOROHOD SPACE AND REPRESENTATIONS OF CàDLàG INFINITELY DIVISIBLE PROCESSES

摘要

Let X_n be independent random elements in the Skorohod space D([0, 1];E) of càdlàg functions taking values in a separable Banach space E. Let S_n =∑_(j =1)~n X_j. We show that if S_n converges in finite dimensional distributions to a càdlàg process, then S_n+y_n converges a.s. pathwise uniformly over [0, 1], for some y_n ∈ D([0, 1];E). This result extends the It?–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 càdlàg 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 càdlàg modifications, which may also be of independent interest.