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Theory of Probability and Mathematical Statistics

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Multi-dimensional additively stationary random functions on convex structures

Authors: O. I. Ponomarenko and Yu. D. Perun
Translated by: V. Zayats
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 74 (2006).
Journal: Theor. Probability and Math. Statist. 74 (2007), 133-146
MSC (2000): Primary 60G10; Secondary 60G57
Published electronically: July 5, 2007
MathSciNet review: 2336784
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Abstract: Some classes of the wide-sense additively stationary generalized random functions taking values in a complex Hilbert space are considered. These random functions are defined on certain types of convex cones and convex sets belonging to a real vector space that can be interpreted as commutative additive semigroups endowed with the identical involution $ *$. Here, stationarity is understood as $ *$-stationarity with respect to a semigroup in the sense of earlier papers by the authors. For the above-mentioned classes of additively stationary random functions, spectral expansions are obtained for both these functions and their correlation functions. Properties of these expansions are studied and the problem of the extension of the described additively stationary functions to wider sets in vector spaces is considered.

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Additional Information

O. I. Ponomarenko
Affiliation: Department of Probability Theory and Mathematical Statistics, Mechanics and Mathematics Faculty, Taras Shevchenko National University, Glushkov Ave., 6, Kyïv 03127, Ukraine

Yu. D. Perun
Affiliation: Auditorship Department, National Bank of Ukraine, Instituts’ka Street, 9, Kyïv 01601, Ukraine

Keywords: Additively stationary random function in a Hilbert space, convex cone, convex set, $\alpha$-boundedness, spectral expansion
Received by editor(s): April 13, 2005
Published electronically: July 5, 2007
Article copyright: © Copyright 2007 American Mathematical Society