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Transactions of the American Mathematical Society

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Stochastic processes with sample paths in reproducing kernel Hilbert spaces


Authors: Milan N. Lukic and Jay H. Beder
Journal: Trans. Amer. Math. Soc. 353 (2001), 3945-3969
MSC (2000): Primary 60G12; Secondary 60B11, 60G15, 28C20, 46E22, 47B32
DOI: https://doi.org/10.1090/S0002-9947-01-02852-5
Published electronically: May 14, 2001
MathSciNet review: 1837215
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Abstract | References | Similar Articles | Additional Information

Abstract:

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 $1$.

Doing away with Driscoll's restrictions, R. Fortet generalized his condition and named it nuclear dominance. He stated a theorem claiming nuclear dominance to be necessary and sufficient for the 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.

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.


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Additional Information

Milan N. Lukic
Affiliation: Department of Mathematics, Viterbo University, 815 South 9th Street, La Crosse, Wisconsin 54601
Email: mnlukic@viterbo.edu

Jay H. Beder
Affiliation: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, Wisconsin 53201-0413
Email: beder@uwm.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02852-5
Keywords: Covariance operator, Gaussian process, nuclear dominance, random element in Hilbert space, reproducing kernel Hilbert space, second order process, zero-one law
Received by editor(s): March 10, 2000
Received by editor(s) in revised form: February 8, 2001
Published electronically: May 14, 2001
Article copyright: © Copyright 2001 American Mathematical Society