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.
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 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
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 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.