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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

Discrete representations of second order random functions. I


Author: O. I. Ponomarenko
Translated by: S. V. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 86 (2012).
Journal: Theor. Probability and Math. Statist. 86 (2013), 183-192
MSC (2010): Primary 60G10, 60G15; Secondary 60G57
Published electronically: August 20, 2013
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Abstract: We study representations of both scalar and vector random functions defined on a set $ T$ in the form of either infinite or finite sums whose terms are scalar functions on $ T$ with random coefficients. The assumptions imposed on the set $ T$ and on properties of random functions are rather general. In particular, we consider the cases where $ T$ is a compact topological space, measurable space with positive measure, or an arbitrary nonempty set. Various examples of such representations are given for specific random functions assuming values in Hilbert spaces. The Karhunen-Loeve type representations are considered in part I of the paper. More general basis type representations are studied in part II.


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Additional Information

O. I. Ponomarenko
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine

DOI: http://dx.doi.org/10.1090/S0094-9000-2013-00897-7
PII: S 0094-9000(2013)00897-7
Keywords: Random functions defined on a compact topological space, generalized random functions assuming values in a Hilbert space, Karhunen--Loeve type representations, basis type representations, Hilbert--Schmidt operators
Received by editor(s): September 8, 2011
Published electronically: August 20, 2013
Article copyright: © Copyright 2013 American Mathematical Society