In K. L. Chung's book, the first exercise of Chapter 5: Find a sequence of random variables (may not be independent!) such that the momentum of the second order goes to zero but it does not satisfies the strong law of large number.
Now, I have reduced this problem to the following one: Find a sequence of random variables $(Y_n)$, which does not converge to zero almost surely, such that $E[Y_n]=0$, and $n Y_n -(n-1) Y_{n-1}$ tends to zero in $L^2$.
Actually, one can not construct a non-a.e. convergence sequence with the momentum of second order going as $o(1/n^2)$. So I think there should be some delicate construction of this problem. Can anyone give some suggestions about that?