(Solution Library) Let (X_i) be i.i.d. with E X_i^2 var;(S_T)=(var;(X_1))(E T) ? If not, is it true in the special case E X_1=0
Question: Let \(\left(X_{i}\right)\) be i.i.d. with \(E X_{i}^{2}<\infty .\) Let \(S_{n}=\sum_{i=1}^{n} X_{i} .\) Let \(T\) be a bounded stopping time. Is it true in general that
\[\operatorname{var}\left(S_{T}\right)=\left(\operatorname{var}\left(X_{1}\right)\right)(E T) ?\]If not, is it true in the special case \(E X_{1}=0 ?\)
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