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I am trying to solve the following exercise from my professor's notes on conditional expectation:

Let $x: \Omega \rightarrow \mathbb{R}^n$, $x \in G(0, Q_x)$, $Q_x = Q_x^T>0$, $y: \Omega \rightarrow \mathbb{R}^k$, $y \in G(0, Q_y)$. Assume that $x$, $y$ are jointly Gaussian random variables and that \begin{equation*} E[\exp(iu^Ty)|F^x] = \exp\left(iu^TCx - \frac{1}{2} u^T \tilde{Q}u\right), \end{equation*} for some $C \in \mathbb{R}^{k \times n}$, $\tilde{Q} = \tilde{Q}^T \geq 0$. Prove that there exists a random variable $w: \Omega \rightarrow \mathbb{R}^k$, such that $x$, $w$ are independent, $w \in G(0, Q_w)$ and $y = Cx + w$

My problem is that I cannot find a way to prove the existence (the inverse is easy to prove). I was thinking to use the property $E[E[\exp(iu^Ty)|F^x]] = E[\exp(iu^Ty)]$ (because the characteristic function would give an expression for $y$) but following this path it gets overly complex.

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Define $w=y-Cx$, then $$ E[\mathrm e^{\mathrm iu^Tw}|\mathcal F^x] =E[\mathrm e^{\mathrm iu^Ty}|\mathcal F^x]\,\mathrm e^{-\mathrm iu^TCx} = \mathrm e^{-u^T \tilde{Q}u/2} $$ is independent of $x$. Thus $w$ is independent of $x$ and centered normal with covariance $\tilde{Q}$.

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  • $\begingroup$ Thank you! Quite simple in the end and similar concept to the proof of the gaussian conditional expectation, but my mind didn't go there because I was thinking of existence theorems. $\endgroup$ – CuriousNik Feb 20 '14 at 10:13

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