# Prove that the following limit is 0

I have to prove this following statement.

Let $(X_n)_{n\in\Bbb N}$ a sequence of independent random variables with $\mathbf E(X_i) = 0$ and $\mathbf{Var}(X_i)=C<\infty$ for all $i \in \Bbb >N$. Let $p >1/2$ and $S_n=X_1+...+X_n$.

Show that $\lim_{n\to \infty}S_n/n^p =0$ in probability.

I think that I should use the central limit theorem, but I am not sure because this way does not lead me to something..

• Hint: Show the convergence in $L^2$. – Did Mar 27 '17 at 9:18