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.
Best Answer
The assertion "$a_1X(t_1)+a_2X(t_2)+\ldots +a_nX(t_n)$ is a jointly gaussian random variable" in your second statement is almost completely correct: $a_1X(t_1)+a_2X(t_2)+\ldots +a_nX(t_n)$ is indeed a gaussian random variable but since it is just one variable all by itself, the adjective "jointly" is not needed.
Definition: A random process $\{X(t) \colon t \in \mathbb T\}$is called a Gaussian random process if all the finite-dimensional distributions of the process are (multivariate) Gaussian distributions.
A more prolix description is that each random variable $X(t), t \in \mathbb T$ is a Gaussian random variable, and for any integer $n \geq 2$ time instants $t_1, t_2, \ldots, t_n \in \mathbb T$, the $n$ random variables $X(t_1)$, $X(t_2)$, $\ldots X(t_n)$, are jointly Gaussian random variables.
Definition: The random variables $X_1, X_2, \ldots, X_n$ are said have a multivariate Gaussian distribution (or they are called jointly Gaussian random variables) if for all choices of real numbers $a_1, a_2, \ldots, a_n$, the random variable $a_1X_1+a_2X_2+\cdots+a_nX_n$ is a Gaussian random variable.
As the OP has noted, this implies that the marginal distributions of the $X_i$ are Gaussian. Note that also that when each $a_i$ is $0$, the sum is also $0$ and we are accepting this constant as a degenerate Gaussian random variable (cf. the extended discussion in the comments following this answer).
Inserting the latter definition into the former, we get the description of Gaussian random process used by the OP. Note that each random variable from a Gaussian random process is indeed a Gaussian random variable. That is, the answer to the OP's question
is an unequivocal Yes.