Generalized stationary random fields with linear regressions - an operator approach

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