Let $X_1,...,X_n$ be $n$ correlated random variables with covariance matrix $\Sigma_n$.
Is simulating the whole system $X_1,...,X_n$ with covariance matrix $\Sigma_n$ and recording the occurances of, say, $X_1,...,X_k$ for some $k\neq n$ the same as simulating only $X_1,...,X_k$ with the submatrix $\Sigma_k$ containing only the covariances between them, thereby ignoring variables upon which they are dependent, like $X_n$ ?
Is this true in general ? What are the conditions that the $X_i$ have to satisfy so that this is true ?
I am asking this question because I have never been exposed rigourously to this subject, and the answers I got (or maybe misunderstood) from my physics class are either unclear or sometimes contradictory.
I was once told that gaussian processes are entirely determined by all the means and all the covariances. I understand that a single normal variable $X$ is determined only by $\mu$ and $\sigma$, but what if there are more of them and they interact with each other ? Does "all the covariances" mean all the covariances of the subsystem or all the covariances of the whole system ? It makes a huge difference in terms of computation time...
My observation is that it seems to be true, at least for normal random variables, but I am still unsure...