Multidimensional weakly stationary random functions on semigroups

Authors:
O. I. Ponomarenko and Yu. D. Perun

Translated by:
V. V. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **73** (2005).

Journal:
Theor. Probability and Math. Statist. **73** (2006), 151-162

MSC (2000):
Primary 60G10, 60G57, 60G15

DOI:
https://doi.org/10.1090/S0094-9000-07-00689-8

Published electronically:
January 17, 2007

MathSciNet review:
2213849

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider some problems in the spectral analysis of weakly stationary Hilbert-valued random functions on involutive semigroups. We obtain spectral representations for such functions and for their correlation functions. These representations are extensions and improvements of the corresponding results proved earlier by the first author and by V. Girardin and R. Senoussi.

**1.**A. I. Ponomarenko,*Banach-space-valued random fields that are homogeneous in the wide sense on semigroups and homogeneous spaces*, Teor. Verojatnost. i Mat. Statist.**7**(1972), 110–121, 165 (Russian, with English summary). MR**0322940****2.**A. I. Ponomarenko,*Infinite-dimensional random fields on semigroups*, Teor. Veroyatnost. i Mat. Statist.**30**(1984), 136–142 (Russian). MR**800839****3.**Valerie Girardin and Rachid Senoussi,*Semigroup stationary processes and spectral representation*, Bernoulli**9**(2003), no. 5, 857–876. MR**2047689**, https://doi.org/10.3150/bj/1066418881**4.**O. Ī. Ponomarenko,*Second-order random linear functionals. I*, Teor. Ĭmovīr. Mat. Stat.**54**(1996), 137–146 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**54**(1997), 143–151. MR**1644594****5.**L. L. Ponomarenko,*Infinite-dimensional stochastic partial differential equations with a multiparameter Brownian motion*, Kibernetika (Kiev)**4**(1976), 98–106 (Russian, with English summary). MR**0488294****6.**R. J. Lindahl and P. H. Maserick,*Positive-definite functions on involution semigroups*, Duke Math. J.**38**(1971), 771–782. MR**291826****7.**Christian Berg, Jens Peter Reus Christensen, and Paul Ressel,*Harmonic analysis on semigroups*, Graduate Texts in Mathematics, vol. 100, Springer-Verlag, New York, 1984. Theory of positive definite and related functions. MR**747302****8.**Helge Glöckner,*Positive definite functions on infinite-dimensional convex cones*, Mem. Amer. Math. Soc.**166**(2003), no. 789, xiv+128. MR**2008256**, https://doi.org/10.1090/memo/0789**9.**A. I. Ponomarenko,*Stochastic integrals with respect to generalized random orthogonal measures in Banach spaces*, Teor. Veroyatnost. i Mat. Statist.**33**(1986), 92–99, 126 (Russian). MR**898202****10.**O. Ī. Ponomarenko,*Integral representation of random functions with values in locally convex spaces*, Teor. Īmovīr. ta Mat. Statist.**46**(1992), 132–141 (Ukrainian, with Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist.**46**(1993), 129–136. MR**1196216****11.**Olivier Perrin and Rachid Senoussi,*Reducing non-stationary stochastic processes to stationarity by a time deformation*, Statist. Probab. Lett.**43**(1999), no. 4, 393–397. MR**1707949**, https://doi.org/10.1016/S0167-7152(98)00278-8**12.**O. Ī. Ponomarenko,*Operator bimeasures and stochastic integrals in normed spaces*, Teor. Īmovīr. ta Mat. Statist.**47**(1992), 129–139 (Ukrainian, with Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist.**47**(1993), 129–137. MR**1272232****13.**N. I. Akhiezer,*The classical moment problem and some related questions in analysis*, Translated by N. Kemmer, Hafner Publishing Co., New York, 1965. MR**0184042****14.**Jiří Michálek,*Locally stationary covariances*, Transactions of the Tenth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Vol. A (Prague, 1986) Reidel, Dordrecht, 1988, pp. 83–103. MR**1136266****15.**A. I. Ponomarenko,*Stochastic Problems of Optimization*, Kiev University, Kiev, 1980. (Russian)

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Additional Information

**O. I. Ponomarenko**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kiev 03127, Ukraine

Email:
probab@univ.kiev.ua

**Yu. D. Perun**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kiev 03127, Ukraine

Email:
perun@bank.gov.ua

DOI:
https://doi.org/10.1090/S0094-9000-07-00689-8

Keywords:
Random functions,
spectral representations,
involutive semigroup,
weakly stationary random functions

Received by editor(s):
December 3, 2004

Published electronically:
January 17, 2007

Article copyright:
© Copyright 2007
American Mathematical Society