Let $X_1,X_2,...$ be iid random variables in $L^1$ and $T$ be a stopping time. Prove
$E(S_T-E(X_i)T)^2=Var(X_i)E(T)$.
I understand the proof when $E(X_i)=0$ but I am having difficulty for the case otherwise. I'm thinking of shifting/centering the random variable $X$ but am unsure as to how to implement this exactly.
As I long I know for centered $X_1$ the proof is along the lines of using optimal stopping, like here.
Define $\xi_i := X_i - \Bbb E [X_i]$. Since $\xi_1$ is centered we have (assuming we understand the centered case) that $$\Bbb E[(S_T - \Bbb E[X_1] T)^2]= \Bbb E [(\sum_{i=1}^T X_i - \sum_{i=1}^T \Bbb E[X_i])^2] = \Bbb E[(\sum_{i=1}^T \xi_i)^2]\\ \overset{(*)}{=} \Bbb V (\xi_1) \Bbb E[T] = \Bbb V (X_i)\Bbb E[T] $$
where in $(*)$ we use the result from here.