This is a continuation of the previous post that can be found here. We discuss the Generalised CLT in this post. We give a sloppy proof of the theorem. Then we end with a series representation theorem for the Stable distributions.
The Generalised Central Limit Theorem
Theorem 9: are iid symmetric random variables with as for some and . Then
where with appropiate constants.`
Motivation for proof: Recall the proof of usual CLT. We assume are iid random variables mean and variance and characteristic function . We use the fact that under finite variance assumption we have
and hence using this we get
We will this idea in our proof. We will leave some of the technical details of the proof. The proof presented here is not rigourous. We primarily focus on the methodology and tricks used in the proof.