Generalized stationary random fields with linear regressions - an operator approach

Existence, $L^2$-stationarity and linearity of conditional expectations $\wwo{X_k}{...,X_{k-2},X_{k-1}}$ of square integrable random sequences $\mathbf{X}=(X_{k})_{k\in\mathbb{Z}}$ satisfying \[ \wwo{X_k}{...,X_{k-2},X_{k-1},X_{k+1},X_{k+2},...}=\sum_{j=1}^\infty b_j(X_{k-j}+X_{k+j}) \] for a real sequence $(b_n)_{n\in\nat}$, is examined. The analysis is reliant upon the use of Laurent and Toeplitz operator techniques.