The problem is
Suppose $P(X_j=j)=P(X_j=-j)=1/2j^{\beta}$ and $P(X_j=0)=1-j^{-\beta}$ where $\beta>0$. Show that if $\beta <1 $ then $S_n/n^{(3-\beta)/2}\Rightarrow c \chi$.
Here is one solution using characteristic function( Let $P(X_j=j)=P(X_j=-j)=1/2j^{\beta}$ and $P(X_j=0)=1-j^{-\beta}$ where $\beta\in(0,1)$, then $S_n/n^{(3-\beta)/2)}\Rightarrow c\chi$ ).
But can we prove this using Lindeberg Feller Theorem?
Letting $X_{n,m} = \frac{X_m}{n^{(3-\beta)/2}}$, it is easy to show that $\lim_{n\rightarrow\infty}\sum\limits^n_{m=1}E[ X_{n,m}^2]\rightarrow \sigma$.
However I can't prove that $$ \lim_{n\rightarrow\infty}\sum\limits^n_{m=1}E[X_{n,m}^2;|X_{n,m}|>\epsilon]\rightarrow 0$$