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..

Thanks in advance

  • $\begingroup$ Is it given that the distributions of the variables are the same? $\endgroup$ – NCh Mar 27 '17 at 8:49
  • $\begingroup$ @NCh No, I do not think so. $\endgroup$ – GYBE Mar 27 '17 at 9:03
  • 4
    $\begingroup$ Hint: Show the convergence in $L^2$. $\endgroup$ – Did Mar 27 '17 at 9:18
  • $\begingroup$ @DavideDima Then you cannot use central limit theorem without additional conditions. $\endgroup$ – NCh Mar 27 '17 at 11:48

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