I have a question regarding the distribution of the sum of a discrete-time stochastic process. That is, if the stochastic process is $(X_1,X_2,X_3,X_4,\ldots)$, what is the distribution of $X_1+X_2+X_3+\ldots$? $X_i$ could be assumed from a discrete or continuous set, whatever is easier to calculate.
I understand that it mainly depends on the distribution of $X_i$ and on the fact if the $X_i$ are correlated, right? If they are independent, the computation is probably relatively straightforward, right? For the case of two variables, it is the convolution of the probability distributions and probably this can be generalized to the case of n variables, does it? But what if they are dependent?
Are there any types of stochastic processes, where the distribution of the sum can be computed numerically or even be given as a closed-form expression?
I really appreciate any hints!