I searched for a proof for the generalisation about the indepence of sums of independent random variables, but I can't find the result anywhere and I've also got quite some difficulty with proving the result myself, can anyone give me some good hints on how to proof this, or a link to the proof of the result itsself? What I'm trying to proof is the following:
Let $X_1, X_2, \dots X_n$ be independent discrete random variables all with the same image, then for all partitions $P_1, P_2, \dots P_j$ of the index set $\{1, 2, \dots, n\}$ the random variables $R_1 = \sum\limits_{i \in P_1} X_i $$,$ $R_2 = \sum\limits_{i \in P_2} X_i $ $,\dots,$ $R_j = \sum\limits_{i \in P_j} X_i $ are independent.