We have a set $X_1, X_2, \ldots, X_n$ of correlated discrete random variable with a given correlation matrix. How can one compute the sum $X_1 + X_2 + \cdots+ X_n$ knowing the probability mass functions of each variable and the correlation matrix?
Can you also please point me to some relevant papers to read ?
Thanks, Bogdan.
You can't. There's more to the joint probability mass function than the marginals and the correlations. For example, consider two random variables $X_1, X_2$, each uniformly distributed on $\{-1,0,1\}$, and with correlation $0$.
They could, for example, be independent, or they could have joint probability mass function $$ \pmatrix{1/6 & 0 & 1/6\cr 0 & 1/3 & 0\cr 1/6 & 0 & 1/6\cr}$$ or $$ \pmatrix{2/9 & 0 & 1/9\cr 0 & 1/9 & 2/9\cr 1/9 & 2/9 & 0\cr}$$ And each of these cases gives a different distribution to $X_1 + X_2$ .