[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.

This entry was posted in Undergraduate and tagged . Bookmark the permalink.

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s