Is sum of two sequences of independent i.i.d random variables independent?

Question

I am looking for a proof that if I have two sequences of i.i.d random variable that are independent $$ (X)^\infty_{n=1} , (Y)^{\infty}_{n=1} $$ then its' sum $ (X+Y)_{k} $ is also independent from $ (X+Y)_{m} $ for $ \forall _{m,l} $

I want to use it in a proof that sum of two independent Poisson processes has independent increments and I guess this theorem is general, am I right ?

Answer 1

Let $f,g$ be the common characteristic function of the two i.i.d sequences. For $k \neq m$ we have $Ee^{it(X_k+Y_k)+is(X_m+Y_m)}=f(t)g(t)f(s)g(s)=Ee^{it(X_k+Y_k)}Ee^{is(X_m+Y_m)}$ and this implies that $X_k+Y_k$ is independent of $X_m+Y_m$.