TITLE: Expected value and quadratic sum
QUESTION [0 upvotes]: Consider a sequence of random variables $(X_k)_{k \in \mathbb{N}} , \ L=\sum_{k=0}^{\infty} v^k X_k$ with $\operatorname{E}(X_k)=0 \forall k$ I want to calculate: $$\operatorname{Var(L)}= \operatorname{E}\left(\sum_{k=0}^{\infty}v^k X_k \right)^2 - 0 \overset{!}{=} \sum_{k=0}^{\infty} v^{2k} \operatorname{Var}(X_k)$$
How do I get to that result?

REPLY [1 votes]: $$\operatorname{Var(L_{n})}= \operatorname{E}\left(\sum_{k=0}^{n}v^k X_k \right)^2{=} \sum_{k=0}^{n} v^{2k} \operatorname{Var}(X_k)$$
because $ v^k X_k$ are also independent :
If I'm allowed to assume that $$\operatorname{Var}(X_k)$$ and $$\operatorname{E}(X_k)$$ are bounded then you can take the limit as n approaches $\infty$
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