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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Generalized stationary random fields with linear regressions - an operator approach


Authors: Wojciech Matysiak and Pawel J. Szablowski
Journal: Trans. Amer. Math. Soc. 360 (2008), 3909-3919
MSC (2000): Primary 60G12; Secondary 60G10, 47B35
Published electronically: February 27, 2008
MathSciNet review: 2386251
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Abstract: Existence, $ L^2$-stationarity and linearity of conditional expectations $ \mathbb{E}[X_k\vert\ldots,X_{k-2},X_{k-1}]$ of square integrable random sequences $ \mathbf{X}= \left( X_{k}\right)_{k\in\mathbb{Z}}$ satisfying

$\displaystyle \mathbb{E}[X_k\vert{\ldots,X_{k-2},X_{k-1},X_{k+1},X_{k+2},\ldots}]=\sum_{j=1} ^\infty b_j\left(X_{k-j}+X_{k+j}\right)$

for a real sequence $ \left(b_n\right)_{n\in\mathbb{N}}$ is examined. The analysis is reliant upon the use of Laurent and Toeplitz operator techniques.


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Additional Information

Wojciech Matysiak
Affiliation: Wydział Matematyki i Nauk Informacyjnych, Politechnika Warszawska, Pl. Politechniki 1, 00-661 Warszawa, Poland – and – Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025
Email: matysiak@mini.pw.edu.pl

Pawel J. Szablowski
Affiliation: Wydział Matematyki i Nauk Informacyjnych, Politechnika Warszawska, Pl. Politechniki 1, 00-661 Warszawa, Poland
Email: pszablowski@elka.pw.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04409-7
PII: S 0002-9947(08)04409-7
Keywords: Linear regressions, stationary random sequences, Laurent operators, Toeplitz operators, harnesses.
Received by editor(s): July 21, 2005
Received by editor(s) in revised form: August 16, 2006
Published electronically: February 27, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.