Generalized stationary random fields with linear regressions - an operator approach

Matysiak, Wojciech; Szabłowski, Paweł

doi:10.1090/S0002-9947-08-04409-7

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.

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