What are the exact conditions to get a multivariate gaussian distribution from multiple single gaussian variables?

Question

I would like to get a precise answer to this question, I can't seem to find a clear answer anywhere.

Moreover, what about this special case :

I have $m$ gaussian random variables $X_i$ ($i=1,...,m$), which are dependent because they are defined by

$$ X_1 = X - a_1\\ X_2 = X - a_2\\ \vdots\\ X_m = X - a_m, $$

where the $a_i$ are real positive constants and $X$ is a random variable following $N(\mu, \sigma^2)$.

So they all have the same variance, but not the same mean. Is their joint distribution a multivariate gaussian? And if not, what can I say about their joint distribution?

Thanks a lot.

Answer 1

In general if $X:\Omega\to\mathbb R^n$ is a random vector with normal distribution where $\mathsf EX=\mu\in\mathbb R^n$ and $\mathsf{Covar}X=\Sigma$ then $AX+v$ where $A^{m\times n}$ is a matrix with entries in $\mathbb R$ and $v\in\mathbb R^m$ also has normal distribution.

This with $\mathsf E(AX+v)=A\mu+v$ and $\mathsf{Cov}(AX+v)=A\Sigma A^T$.

Now apply this for $n=1$ and $A^{m\times1}=(1,\dots,1)^T$ and $v=(-a_1,\dots,-a_m)^T$ and you are ready.