For , define the random variable

Let . Prove that as ,

a) the distribution of converges to for some real number ,

b) but converges to 2.

(*Credits:* I learnt of this problem from Persi Diaconis in my probability class.)

For each , let be the **simple truncation** of , i.e. , or equivalently

It is easy to calculate the following:

If we let , then

Let us check that **Lyapounov’s condition** holds for :

Since , as , the denominator of the above goes to infinity, which means that the entire RHS goes to zero, i.e. Lyapounov’s condition holds. Hence, we can use **Lyapounov’s Central Limit Theorem** to conclude that

(Here, means “converges in distribution”.) Next, note that

By the **first Borel-Cantelli lemma**, we conclude that . From this, we can further conclude that

where .

It is straightforward to show that converges to 2:

**Done!**

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