# How to check the sequence of independent random variables satisfy Lindeberg condition?

The sequence of independent random variables has the following distribution. $$\alpha>0\,, \quad P(X_n=n)=P(X_n=-n)=\frac{1}{6n^{2(\alpha-1)}}\,,\quad P(X_n=0)=1-\frac{1}{3n^{2(\alpha-1)}}$$ I need to prove that the Lindeberg condition holds if and only if $$\alpha<\frac{3}{2}$$. In the case of above distribution, I turn the problem into proving $$\frac{\sum_{k=\min\left(a:a\geq \epsilon \sqrt{\sum_{i=1}^{n}\operatorname{Var}(X_i)}\right)}^{n}\frac{1}{3}k^{4-2\alpha}}{\sum_{k=1}^{n}\frac{1}{3}k^{4-\alpha}} \rightarrow 0$$ How to get the result?

Lindeberg condition: $$\{ X_n,n\ge1 \} \text{is a sequence of independent random variable.}\\EX_k=\mu_k \quad \operatorname{Var}X_k=\sigma_k^2 \quad B_n=\sum_{k=1}^{n} \sigma_k^2 \\ \forall \epsilon > 0 \quad \frac{1}{B_n}\sum_{k=1}^{n}\int_{|x-\mu_k|>\epsilon\sqrt{B_n}}(x-\mu_k)^2dF_k(x) \rightarrow 0 \quad (n\rightarrow \infty)$$

We have to check that $$R_n:=\frac{\sum_{k=\min\left(a:a\geq \epsilon \sqrt{\sum_{i=1}^{n}\operatorname{Var}(X_i)}\right)}^{n} k^{4-2\alpha}}{\sum_{k=1}^{n} k^{4-2\alpha}} \rightarrow 0.$$ Let $$B_n=\sum_{i=1}^{n}\operatorname{Var}(X_i)$$. If $$\alpha\gt 5/2$$, then $$B_n$$ converges to a positive number and $$R_n$$ will converge to $$1$$.

If $$\alpha =5/2$$, then $$R_n$$ behaves as $$R'_n:=\frac 1{\log n}\sum_{k=\lfloor c \log n\rfloor }^nk^{-1}$$ which goes also to $$1$$.

Now, assume that $$\alpha\lt 5/2.$$ Then one can find constants $$c_1$$ and $$c_2$$ depending only on $$\alpha$$ such that $$c_1n^{5-2\alpha}\leqslant B_n\leqslant c_2n^{5-2\alpha}$$ and a similar bound holds for $$B_n$$. As a consequence, the summation set in the denominator is empty for $$n$$ large enough in the case $$\alpha\lt 3/2$$.

If $$\alpha \gt 3/2$$, one gets
$$\sum_{k=\min\left(a:a\geq \epsilon \sqrt{\sum_{i=1}^{n}\operatorname{Var}(X_i)}\right)}^{n} k^{4-2\alpha}\geqslant \sum_{k=\lfloor Kn\rfloor}^nk^{4-2\alpha}\geqslant K'n^{5-2\alpha}$$ for some positive $$K'$$ hence $$R_n$$ does not go to zero.