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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Norm convergent expansions for Gaussian processes in Banach spaces.


Authors: Naresh C. Jain and G. Kallianpur
Journal: Proc. Amer. Math. Soc. 25 (1970), 890-895
MSC: Primary 60.50
MathSciNet review: 0266304
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Abstract: Several authors have recently shown that Brownian motion with continuous paths on $ [0,1]$ can be expanded into a uniformly convergent (a.s.) orthogonal series in terms of a given complete orthonormal system (CONS) in its reproducing kernel Hilbert space (RKHS). In an earlier paper we generalized this result to Gaussian processes with continuous paths. Here we obtain such expansions for a Gaussian random variable taking values in an arbitrary separable Banach space. A related problem is also considered in which starting from a separable Hilbert space $ H$ with a measurable norm $ \vert\vert \cdot \vert{\vert _1}$ defined on it, it is shown that the corresponding abstract Wiener process has a $ \vert\vert\cdot\vert{\vert _1}$-convergent orthogonal expansion in terms of a CONS chosen from $ H$.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0266304-1
Keywords: Uniformly convergent a.s., orthogonal expansion, Gaussian process
Article copyright: © Copyright 1970 American Mathematical Society