This is a continuation of the previous post that can be found here. We will discuss some of the properties of the Stable distributions in this post.
Theorem 2. is the limit of for some iid sequence and sequence and if and only if has a stable law.
Remark: This kind of explains why Stable laws are called Stable!
Proof: The if part follows by taking to be stable itself. We focus on the only if part.
Let . Fix a . Let
Now consider . Observe that
where ‘s are iid copies of . Let and let
where and .
Now . and . If we can show that and , then this will ensure
Note that and depends only on . Hence the law of is stable.