I am given a proposition that states random vectors $(X,Y)$ form a joint Gaussian distribution if and only if $X$ is Gaussian and $Y|X$ is Gaussian.
Using this result it states that if for a random vector $(X,Y,Z)$ the marginal $Z$ is Gaussian, conditional $Y|Z$ is Gaussian and $X|Y,Z$ is Gaussian then $(X,Y,Z)$ is Gaussian.
At first glance it looks like a simple application of the proposition twice but I found that the order is not correct.
So if $Z$ is Guassian and $Y|Z$ is Gaussian then by the proposition, $(Z,Y)$ is Gaussian not $(Y,Z)$. Then we have $(Z,Y)$ as $X$ in the proposition and $(Y,Z)$ is $X$ in the proposition so we should have $(Y,Z,X)$ is a joint Gaussian distribution.
I feel like the order is all messed up here. There is no guarantee of exchangeability here and shouldn't the order in the conditional vectors matter too? Is that to be assumed or am I missing something here?