I'm working the question below: $X_1, X_2, \dots $ are uncorrelated random variables with $E(X_i) = \mu_i $ and $\operatorname{var}(X_i)/i \rightarrow 0$ as $i \rightarrow \infty$. Now, let $S_n = X_1 + X_2 +\dots + X_n$ and $\nu_n = E(S_n)/n$.
The goal is to show that as $n \rightarrow \infty$ then $S_n/n - \nu_n\rightarrow 0$ in mean squared and in probability.
Could you please explain what do we mean by convergence in mean squared? Also can you guide me on solving this question?
I appreciate your help.