Stable Distributions: Generalised CLT and Series Representation Theorem

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: {X_1,X_2,\ldots} are iid symmetric random variables with {P(X_1>\lambda) \sim k\lambda^{-\alpha}} as {\lambda \rightarrow\infty} for some {k>0} and {\alpha \in (0,2)}. Then

\displaystyle \frac{X_1+X_2+\cdots+X_n}{n^{1/\alpha}} \stackrel{d}{\rightarrow} Y

where {Y\sim S\alpha S} with appropiate constants.`

Motivation for proof: Recall the proof of usual CLT. We assume {X_i} are iid random variables mean {0} and variance {\sigma^2} and characteristic function {\phi}. We use the fact that under finite variance assumption we have

\displaystyle \frac{1-\phi(t)}{t^2} \rightarrow \frac{\sigma^2}{2}

and hence using this we get

\displaystyle \begin{aligned} E\left[e^{i\frac{t}{\sqrt{n}}(X_1+X_2+\cdots+X_n)}\right] = \left[E\left(e^{i\frac{t}{\sqrt{n}}X_1}\right)\right]^n & = \left[\phi\left(\frac{t}{\sqrt{n}}\right)\right]^n \\ & = \left[1-\left(1-\phi\left(\frac{t}{\sqrt{n}}\right)\right)\right]^n \\ & \rightarrow \exp(-t^2\sigma^2/2) \end{aligned}

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.

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Stable Distributions: Properties

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. Y is the limit of \displaystyle\frac{X_1+X_2+\cdots+X_n-b_n}{a_n} for some iid sequence X_i and sequence a_n >0 and b_n if and only if Y has a stable law.

Remark: This kind of explains why Stable laws are called Stable!

Proof: The if part follows by taking X_i to be stable itself. We focus on the only if part.

Let \displaystyle Z_n=\frac{X_1+X_2+\cdots+X_n-b_n}{a_n}. Fix a k\in \mathbb{N}. Let

\displaystyle S_n^{j}= \sum_{i=1}^n X_{(j-1)n+i}= X_{(j-1)n+1}+X_{(j-1)n+2}+\cdots+X_{jn} \ \ \mbox{for} \ (1\le j \le k)

Now consider Z_{nk}. Observe that

\displaystyle Z_{nk} = \frac{S_n^1+S_n^2+\cdots+S_n^k-b_{nk}}{a_{nk}}

\displaystyle \implies \frac{a_{nk}Z_{nk}}{a_n} = \frac{S_n^1-b_n}{a_n}+\frac{S_n^2-b_n}{a_n}+\cdots+\frac{S_n^k-b_n}{a_n}+\frac{kb_n-b_{nk}}{a_n}

As n\to \infty,

\displaystyle \frac{S_n^1-b_n}{a_n}+\frac{S_n^2-b_n}{a_n}+\cdots+\frac{S_n^k-b_n}{a_n} \stackrel{d}{\to} Y_1+Y_2+\cdots+Y_k

where Y_i‘s are iid copies of Y. Let Z_{nk}=W_n and let

\displaystyle \begin{aligned} W_n' & :=\frac{a_{nk}Z_{nk}}{a_n}-\frac{kb_n-b_{nk}}{a_n} \\ & = \alpha_nW_n+\beta_n \end{aligned}

where \displaystyle\alpha_n:=\frac{a_{nk}}{a_n} and \displaystyle\beta_n:=-\frac{kb_n-b_{nk}}{a_n}.

Now W_n \stackrel{d}{\to} Y. and W_n'\stackrel{d}{\to} Y_1+Y_2+\cdots+Y_k. If we can show that \alpha_n \to \alpha and \beta_n \to \beta, then this will ensure

\alpha Y+\beta \stackrel{d}{=} Y_1+Y_2+\cdots+Y_k

Note that \alpha and \beta depends only on k. Hence the law of Y is stable.

\square

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