Conditional Expectation for Jointly Gaussian Random Variables is Affine

Let $r$ be a natural number and let $(X,U_1,\dots,U_r)$ be a Gaussian random vector on a probability space $(\Omega,\mathcal{F},P)$. Then there exist real numbers $\beta_0,\beta_1,\dots,\beta_r$ such that the random variable

has the following properties:

1. (Conditional expectation) $Y$ is a conditional expectation of $X$ given the generated $\sigma$-algebra $\sigma(U_1,\dots,U_r)$; consequently, in the notation of Conditional Expectation of a Square-Integrable Random Variable,

2. (Gaussian residual) $X-Y$ is a Gaussian random variable with $\mathbb{E}[X-Y]=0$.

3. (Orthogonality) $X-Y$ is uncorrelated with $U_k$ for every $1\le k\le r$.

4. (Independence of the residual) The $\sigma$-algebras $\sigma(X-Y)$ and $\sigma(U_1,\dots,U_r)$ are independent.