This is a continuation of the previous post that can be foundĀ 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

As ,

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.