Conditional Expectation Given Countably Many Jointly Gaussian Observations

Let $X$ and $U_k$ ($k\in\mathbb{N}$) be random variables on a probability space $(\Omega,\mathcal{F},P)$ such that the family $(X,U_1,U_2,\dots)$ is jointly Gaussian. Write, with the generated $\sigma$-algebras,

Then:

1. (Filtration structure) The sequence $(\mathcal{G}_r)_{r\in\mathbb{N}}$ is nondecreasing and $\sigma\bigl(\bigcup_{r\in\mathbb{N}}\mathcal{G}_r\bigr)=\mathcal{G}$.

2. (Affine finite-horizon estimates) For every $r\in\mathbb{N}$ there exist real numbers $\beta_{r,0},\beta_{r,1},\dots,\beta_{r,r}$ such that

is a conditional expectation of $X$ given $\mathcal{G}_r$.

3. (Mean-square convergence to the full conditional expectation) For every choice of conditional expectations $Y_r$ of $X$ given $\mathcal{G}_r$ as in part 2 and every conditional expectation $Y$ of $X$ given $\mathcal{G}$, the mean-square distances satisfy $\lVert Y_r-Y\rVert_{2}\to0$. In particular, every conditional expectation of $X$ given $\mathcal{G}$ is a mean-square limit of affine combinations of finitely many of the observations $U_k$.

4. (Orthogonality in the limit) For every conditional expectation $Y$ of $X$ given $\mathcal{G}$ and every $k\in\mathbb{N}$, the residual $X-Y$ is uncorrelated with $U_k$.