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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!

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    Unclear. Are you thinking of the process $Z_n = \sum_{k=1}^n X_k$ ? Or the sum of the most recent $M$ values ? ($Z_n = \sum_{k={n-M+1}}^n X_k$ ? Furthermore, $Z-n$ is itself a stochastic process (non stationary, at least in the first case), when you speak of its distribution, are you refering to the distribution of $Z_n$ for a single $n$?2012-08-16
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    Thanks for your comment! I am thinking of the process $(X_1, X_2, ...)$ and of the distribution of $Z_n=\sum_{k=1}^n X_k$ for a certain fixed $n$. I don't see the difference in computation of $Z_n=\sum_{k=1}^{n}X_k$ and of $Z_n=\sum_{k=n-M+1}^n X_k$. Is there any? In both cases it is the computation of distribution of the sum of $n$ (possibly depenent) random variables, right?2012-08-16
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    Have you ever considered the discrete-time ito integral (stochastic calculus)? I am on a more general problem that i want to evaluate the r.v. $f(X_1)+ \cdots + f(X_N)$, where $N$ is also an r.v.. And I know that the continuous analogous version is solvable, while i am finding a way to obtain the discrete one...2015-09-29

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