## [Hints] Central Limit Theorem: Strange Result!

For $n \in \mathbb{N}$, define the random variable

$X_n = \begin{cases} \pm 1 &\text{each with probability } \frac{1}{2}\left( 1 - \frac{1}{n^2} \right), \\ \pm n^2 &\text{with probability } \frac{1}{2n^2}. \end{cases}$

Let $S_n = \displaystyle\sum_{k = 1}^n X_k$. Prove that as $n \rightarrow \infty$,

a) the distribution of $\displaystyle\frac{S_n}{\sqrt{n}}$ converges to $\mathcal{N}(0, a)$ for some real number $a \neq 2$,

b) but $\text{Var} \displaystyle\frac{S_n}{\sqrt{n}}$ converges to 2.

Hint: Simple truncation. Let $Y_n = X_n 1_{\{X_n < n \}}$. Use Lyapounov’s Central Limit Theorem and Borel-Cantelli’s 1st Lemma.