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$(X_k)$ are mutually independent random variables and $S_n=\sum^{n}_{k=1}X_k$. $X_k$ has the following distribution: $P(X_k=k^a)=P(X_k=-k^a)=\frac{1}{2}k^{-b}$, $P(X_k=0)=1-k^{-b}$. If $0\leq b<min(1,2a+1)$, show that there is a real sequence $b_n$ such that $b_n^{-1}S_n\to Z$ in distribution for some r.v. $Z$.

I can think of $b_n=(\sum^n_{k=1}Var(X_k))^{1/2}$ and $Z$ is $N(0,1)$.

I tried to show that in this case $X_n$ satisfies Linderberg condition. However, Linderberg condition is extremely hard to verify. I tried so hard and still can not get an idea.

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