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

• Andreas Basse - O'Conno ^[1] ; Jan Rosiński ^[2]
  1. [1] Aarhus University and University of Tennessee
  2. [2] University of Tennessee
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 6, 2013, págs. 4317-4341
• Idioma: inglés
• DOI: 10.1214/12-AOP783
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• Resumen

  • Let Xn be independent random elements in the Skorohod space D([0,1];E) of càdlàg functions taking values in a separable Banach space E. Let Sn=∑nj=1Xj. We show that if Sn converges in finite dimensional distributions to a càdlàg process, then Sn+yn converges a.s. pathwise uniformly over [0,1], for some yn∈D([0,1];E). This result extends the Itô–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 J1-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.