Let $X$ and $Y$ be of the same dimension and jointly normal. Find the distribution of $X+Y$.

Question

Let $X$ and $Y$ be of the same dimension and jointly normal. Find the distribution of $X+Y$.

Can we start off by saying that if $X$ and $Y$ are jointly normal, then $X$ and $Y$ are normal as well?, i.e.: $$X=AZ_X+\mu_X$$ $$Y=BZ_Y+\mu_Y$$

Answer 1

I don't understand the method you're using up there, but here's how I would pursue this problem.

Hint. Suppose $$\begin{bmatrix} \mathbf{X} \\ \mathbf{Y} \end{bmatrix}\sim\mathcal{N}_{2p}(\boldsymbol{\mu}, \mathbf{V})$$ ($p$ being the dimension of both $\mathbf{X}$ and $\mathbf{Y}$).

A very well-known theorem is the following:

Theorem. Any subvector of a multivariate normal vector is also (multivariate) normal.

It follows immediately from this theorem that $\mathbf{X}$ and $\mathbf{Y}$ must both be normal.

Sketch of proof. Let's suppose $\mathbf{W} \sim \mathcal{N}_k(\boldsymbol{\mu}_{\mathbf{W}}, \mathbf{V}_\mathbf{W})$. Perform induction on the size of the subvector of $\mathbf{W}$, say $m \leq k$.

If we have a subvector of $\mathbf{W}$ with $m = 1$, by definition of a multivariate normal, every subvector of length $1$ of $\mathbf{W}$ must be multivariate normal.

Now try completing this proof.