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Suppose $(x_1, x_2, x_3)\sim N(\mu, \Sigma)$ where $\mu\in\mathbb{R}^3$ and $\Sigma$ is a $3\times 3$ covariance matrix, are the variables $A = x_1 + x_2$ and $B = x_2 + x_3$ necessarily distributed according to a multivariate normal distribution? That is, do there exist $\nu\in\mathbb{R}^2$ and a $2\times 2$ covariance matrix $M$ such that $(A, B) \sim N(\nu, M)$

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Yes and here is how to see it. Write $A$ and $B$ as

$$\begin{bmatrix} A \\B \end{bmatrix}=\begin{bmatrix} 1 & 1 & 0 \\0 & 1 &1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$

and exploit the fact that linear combinations of multivariate normal variables are themselves normal.

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  • $\begingroup$ If you want to do it explicitly I suggest you take a look at how to transform densities and/or the concept of moment generating functions. $\endgroup$ – Martin Schmelzer Sep 21 '15 at 10:59

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