I don't have any basis on multivariate gaussian distributions and amid one exercise I was solving about a non-directly related topic, the following question came to my mind:
Let's say we have some random variables $X_1,\dots,X_n$ such that each $X_i$ follows a gaussian distribution of some kind. Furthermore, assume that $X_1,\dots,X_n$ are independent. Then, is it true that $(X_1,\dots,X_n)$ also follows a gaussian distribution?
Also, an aditional question: in the case that the result is true, is the same statement also true if we don't require that $X_1,\dots,X_n$ are independent?
I am not looking for an elaborate explanation of this facts, just a yes/no so that I can proceed with solving my exercise (again, multivariate gaussian distribution is not the main topic of the exercise and I don't have any basis on it).