Generalized stationary random fields with linear regressions - an operator approach
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- by Wojciech Matysiak and Paweł J. Szabłowski PDF
- Trans. Amer. Math. Soc. 360 (2008), 3909-3919 Request permission
Abstract:
Existence, $L^2$-stationarity and linearity of conditional expectations $\mathbb {E}[X_k|\ldots ,X_{k-2},X_{k-1}]$ of square integrable random sequences $\mathbf {X}= \left ( X_{k}\right )_{k\in \mathbb {Z}}$ satisfying \[ \mathbb {E}[X_k|{\ldots ,X_{k-2},X_{k-1},X_{k+1},X_{k+2},\ldots }]=\sum _{j=1} ^\infty b_j\left (X_{k-j}+X_{k+j}\right )\] 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.References
- Albrecht Böttcher and Bernd Silbermann, Introduction to large truncated Toeplitz matrices, Universitext, Springer-Verlag, New York, 1999. MR 1724795, DOI 10.1007/978-1-4612-1426-7
- Wlodzimierz Bryc, Stationary Markov chains with linear regressions, Stochastic Process. Appl. 93 (2001), no. 2, 339–348. MR 1828779, DOI 10.1016/S0304-4149(00)00094-6
- Włodzimierz Bryc, Stationary random fields with linear regressions, Ann. Probab. 29 (2001), no. 1, 504–519. MR 1825162, DOI 10.1214/aop/1008956342
- Włodzimierz Bryc, Wojciech Matysiak, and Jacek Wesołowski, Quadratic harnesses, $q$-commutations, and orthogonal martingale polynomials, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5449–5483. MR 2327037, DOI 10.1090/S0002-9947-07-04194-3
- Włodzimierz Bryc and Jacek Wesołowski, Conditional moments of $q$-Meixner processes, Probab. Theory Related Fields 131 (2005), no. 3, 415–441. MR 2123251, DOI 10.1007/s00440-004-0379-2
- J. M. Hammersley, Harnesses, Proc. Fifth Berkeley Sympos. Mathematical Statistics and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 89–117. MR 0224144
- J. F. C. Kingman, Random variables with unsymmetrical linear regressions, Math. Proc. Cambridge Philos. Soc. 98 (1985), no. 2, 355–365. MR 795900, DOI 10.1017/S0305004100063520
- J. F. C. Kingman, The construction of infinite collections of random variables with linear regressions, Adv. in Appl. Probab. suppl. (1986), 73–85. MR 868509
- Roger Mansuy and Marc Yor, Harnesses, Lévy bridges and Monsieur Jourdain, Stochastic Process. Appl. 115 (2005), no. 2, 329–338. MR 2111197, DOI 10.1016/j.spa.2004.09.001
- Wojciech Matysiak and PawełJ. Szabłowski, A few remarks on Bryc’s paper on random fields with linear regressions, Ann. Probab. 30 (2002), no. 3, 1486–1491. MR 1920274, DOI 10.1214/aop/1029867134
- —, Bryc’s random fields: The existence and distributions analysis, Preprint, 2003.
- Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
- Otto Toeplitz, Zur Theorie der quadratischen und bilinearen Formen von unendlichvielen Veränderlichen, Math. Ann. 70 (1911), no. 3, 351–376 (German). MR 1511625, DOI 10.1007/BF01564502
- David Williams, Some basic theorems on harnesses, Stochastic analysis (a tribute to the memory of Rollo Davidson), Wiley, London, 1973, pp. 349–363. MR 0362565
- David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. MR 1155402, DOI 10.1017/CBO9780511813658
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
- Paweł J. Szabłowski
- Affiliation: Wydział Matematyki i Nauk Informacyjnych, Politechnika Warszawska, Pl. Politechniki 1, 00-661 Warszawa, Poland
- Email: pszablowski@elka.pw.edu.pl
- Received by editor(s): July 21, 2005
- Received by editor(s) in revised form: August 16, 2006
- Published electronically: February 27, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 3909-3919
- MSC (2000): Primary 60G12; Secondary 60G10, 47B35
- DOI: https://doi.org/10.1090/S0002-9947-08-04409-7
- MathSciNet review: 2386251