Consider a sequence $(X_n)_n$ of random variables. I'm interested in conditions under which, for each $\epsilon > 0$,
\begin{align} \lim_{n \longrightarrow \infty} n P( |X_n-X| > \epsilon \sqrt{n} ) = 0. \end{align}
In particular, I wonder if the result will hold if $\lim\limits_{n \longrightarrow \infty} E[ |X_n-X|^2] = k,$ for some (non random) $k$. The 'standard' idea of using Chebyshev's inequality doesn't help here.
Do you have any suggestions on how to proceed? Thanks alot!