If $X_1, X_2,... $ is iid with $\mathbb{E}(X_i^2) < \infty$ and $\mathbb{E}(X_i)=\mu$, and if $S_0=0$ and $S_n = X_1 + ··· + X_n$, for $n= 1, 2,...,$
what is the minimum mean squared error predictor of $S_{n+1}$ in terms of $S_1,...,S_n$?
Now I know that the random variable $f(X_1,\ldots,X_n)$ that minimises $ \mathbb{E}(X_{n+1} - f(X_1,\ldots,X_n))^2$ is $\mathbb{E}(X_{n+1} \mid X_1,\ldots,X_n)$. Thus the function we require is $\mathbb{E}(S_{n+1} \mid S_n,\ldots,S_1) = \mathbb{E}(S_{n+1}\mid S_n)$. Now since $S_{n+1} = S_n+X_{n+1}$ we know that $\mathbb{E}(S_{n+1}\mid S_{n}) = S_n + \mathbb{E}(X_{n+1}) = S_n + \mu$ and so the minimum mean squared error predictor is $S_n+\mu$ is this correct? I'd appreciate any help thanks!