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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Measurable approximation of a second-order process


Author: Jelena B. Gill
Journal: Proc. Amer. Math. Soc. 100 (1987), 535-542
MSC: Primary 60G12
MathSciNet review: 891160
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Abstract: An important problem, having practical as well as theoretical significance, is the problem of determining whether or not a stochastic process has a measurable modification or can be approximated (in the mean square sense) by a second-order process admitting such a modification. It is known [1] that a second-order process has a measurable modification if and only if its covariance function is measurable and its linear space is separable. However, a problem still open is whether sufficient conditions for the existence of a measurable modification can be given in terms of the covariance function only, so that they have a pure analytical form. In this paper we formulate one set of such conditions (Theorem 2, Corollary 2.2) and from that derive a result giving conditions under which a second-order process can be uniformly approximated (in the mean square sense) by a measurable second-order process (Theorem 3).


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DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0891160-0
PII: S 0002-9939(1987)0891160-0
Keywords: Second-order stochastic process, approximate continuity, covariance function, measurable modification, mean square oscillation, orthogonal dimension
Article copyright: © Copyright 1987 American Mathematical Society