Say I have the expected value of a sum of weakly dependent Normal random variables of the form $\mathbb{E}\left[\sum_{n=1}^\infty a^n X_n\right]$, where $0<a<1$. I was wondering under what conditions this sum might converge and what the expectation then is?
I believe that the sum in the expectation converges absolutely by monotone and dominated convergence if something like $\sup_n \mathbb{E}|X_n|=M<\infty$ holds true (when would this hold for our $X_n$?).
In summary, do you have an idea about under what conditions such a sum converges, and why? Or could you point me at useful references?