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

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.

The fact that and follows from the well known convergence of types theorem which we state below.

**Theorem 3.** *(Convergence of types) *If and there are constants and so that . If are non degenerate distributions, then there are constants and so that and .

**Proof:** The proof involves analysis arguments using characteristic functions. We will skip it. Interested readers may look into Durret[1] for proof.

So far we have introduced stable distributions in the abstract sense. Except for $\alpha=1$ (Cauchy) and $\alpha=2$ (Normal) we dont know if Stable distributions exist at all for other $\alpha$’s. Now we will show the existence through characteristic functions. We will now focus only on the Symmetric stable distributions. Let us assume

where iid where is symmetric.

**Theorem 4. **The characteristic function of is given by .

**Comment:** I didn’t find a simple proof of this fact in books. I asked few professors in our institute, they were also not aware any simple proof. The following proof is my own, largely inspired by Avi levy‘s idea. Well, I do not claim that we are first to discover it!

**Proof:** (Sayan and Avi Levy) Let be the characteristic function. is real and even function. Observe that from relation we have

Note that if we put and in , then

Thus is non negative. Hence has a unique -th root for all positive integers . We will now use it.

Observe that for all we have

Suppose (to be justified later). As . We have

whenver is rational. Now since is continuous and holds on a dense set of . We have

Now we only have to justify the existence of constant . Note that and . If , then for all , which forces by continuity of which is a contradiction. Thus and can be taken as .

Finally we show that symmetric stable distributions actually exists by showing that is a characteristic function of some random variable for .

**Theorem 6:** where is a characteristic function.

**Proof:** The case is settled by normal distribution. So let us assume . Note that for any and $latex|x|\le 1$ we have

where .

Let where

Since we have and . Note that is a characteristic function, and hence (for all ) is a characteristic function. Therefpre, is characteristic function as it is a convex combination of characteristic functions. Thus

For we have

The last one is true because as we have . Hence

Since , we have . Since is continuous at . Using Levy’s continuity theorem we get that is a characteristic function. Hence is a characteristic function.

We will now use the notation to denote a symmetric stable random variable with characteristic function We will drop the parameter from the notation when it is of no interest. From now on we assume . So we do not consider the normal case any more.

**Theorem 7:** , then as where .

**Proof:** The proof is out of scope of our discussion. I will probably add it (if required) later. Readers may look into Samorodnitsky and Taqqu[5] for proof.

**Theorem 8:** , then for all and .

**Proof:** By Theorem 7 we infer that there exist a constant such that . Let .

Using the tail asymptotics of random variable we have

Hence the last sum diverges implying $E|X|^\alpha = \infty$.

We end this section with an application of Theorem 8.

**Application:** are iid random variables with . It is well known via Marcinkiewics-Zygmund law that

We wish to know whether there exist a constant small enough such that

Assume there exist such a . Without loss of generality assume . Set . Observe that . If we consider distribution as the law of ‘s. Then by Theorem 8. But by definition of distribution

Hence we get a contradiction. Thus there exists no such .

In the next post we will look at The Generalised CLT which makes stable distributions so famous.