If $X_1, ..., X_n$ is an iid sequence where $P(X_i = \sqrt{i}) = 1/2$ and $P(X_i = -\sqrt{i}) = 1/2$ for $i = 1,...,n$, how do I verify the Lindeberg condition?
I know that $E(X_i) = 0$ and $Var(X_i) = i$, and hence $S_n = Var(\sum{X_i}) = \frac{n(n+1)}{2}$. But I am getting stuck trying to prove that, as $n \rightarrow \infty$:
$$\frac{1}{S_n^2}\sum_{i = 1}^n E\left(X_i^2I\{|X_i| > \epsilon S_n\}\right) \rightarrow 0$$
A possible hint:
You have $S_n^2=\sum\limits_{k=1}^n\operatorname{Var}( X_k)=\frac{n(n+1)}{2}$.
Now $|X_k|=\sqrt k\le \sqrt n$ almost surely for every $k=1,\ldots,n$.
Observe that $\frac{\sqrt n}{S_n}\to 0$ as $n\to \infty$. So given $\varepsilon>0$, you have $|X_k|<\varepsilon S_n$ almost surely for every $k=1,\ldots,n$ where $n$ is large. In other words, $P\left[|X_k|>\varepsilon S_n\right]= 0$ for all $k=1,\ldots,n$ and for large enough $n$.