If I have a vector $X=(x_1,...,x_n)$ with all $x_i$ normal variables, and I have all the correct conditionals, does this imply $X$ is a multivariate normal?
For example, in two dimensions: Does the fact that $x_1$ and $x_2$ are normal, and $x_1|x_2$ and $x_2|x_1$ are normal with the correct $\mu,\sigma^2$ for a multivariate normal, imply that $X=(x_1,x_2)$ is a multivariate normal?
I tried to fiddle around with the resulting distribution but got nowhere.
Thanks.