Are marginal distributions of the random variables comprising a jointly gaussian random vector always Gaussian?
This stems from my confusion with the Central Limit Theorem which loosely states that sum of a sufficiently large number of independent random variables tends to be normal under mild constraints irrespective of the distributions of each random variable.
The second statement is that of a Gaussia Random process in time, which states that "for any number n of samples, any sampling times $t_1,t_2\ldots ,t_n$, and any scalar constants $a_1,a_2\ldots a_n$, the linear combination $a_1X(t_1)+a_2X(t_2)+\ldots +a_nX(t_n)$ is a jointly gaussian random variable."
Now, for Jointly gaussian random variables $X_1,X_2,\ldots,X_n$, any linear combination of these random variables is a gaussian random variable. Then, for all but one scalar coefficients $a_i$ set to zero, the resulting random variable would still be a Gaussian and hence, the marginal $X_i$ would be gaussian.
But, CLT states that it could be of any distribution!
I am confused with these two lines of thought. Please remedy my confusion.