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Generalized stationary random fields with linear regressions - an operator approach

Author(s): Wojciech Matysiak; Pawel J. Szablowski
Journal: Trans. Amer. Math. Soc. 360 (2008), 3909-3919.
MSC (2000): Primary 60G12; Secondary 60G10, 47B35
Posted: February 27, 2008
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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: Wydzial 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: Wydzial Matematyki i Nauk Informacyjnych, Politechnika Warszawska, Pl. Politechniki 1, 00-661 Warszawa, Poland
Email: pszablowski@elka.pw.edu.pl

DOI: 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.
Posted: February 27, 2008
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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