Stochastic processes with sample paths in reproducing kernel Hilbert spaces

• Published in: Transactions of the American Mathematical Society 2001-10, Vol.353 (10), p.3945-3969
• Main Authors: Milan N. Lukic, Jay H. Beder
• Format: Article
• Language: English
• Subjects: Covariance; Hilbert spaces; Mathematical functions; Mathematical theorems; Probabilities; Radon; Random variables; Separable spaces; Stochastic processes; Trajectories
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-01-02852-5

A theorem of M. F. Driscoll says that, under certain restrictions, the probability that a given Gaussian process has its sample paths almost surely in a given reproducing kernel Hilbert space (RKHS) is either 0 or 1. Driscoll also found a necessary and sufficient condition for that probability to be 1. \par Doing away with Driscoll's restrictions, R. Fortet generalized his condition and named it {\em nuclear dominance}. He stated a theorem claiming nuclear dominance to be necessary and sufficient for the {\em existence\/} of a process (not necessarily Gaussian) having its sample paths in a given RKHS. This theorem -- specifically the necessity of the condition -- turns out to be incorrect, as we will show via counterexamples. On the other hand, a weaker sufficient condition is available. \par Using Fortet's tools along with some new ones, we correct Fortet's theorem and then find the generalization of Driscoll's result. The key idea is that of a random element in a RKHS whose values are sample paths of a stochastic process. As in Fortet's work, we make almost no assumptions about the reproducing kernels we use, and we demonstrate the extent to which one may dispense with the Gaussian assumption.