Stable Distributions: An introduction

In this series of posts I will introduce the stable distributions and discuss some of its properties. The theorems and proofs presented here are written in a lucid manner, so that it can be accessible to most of the readers. I tried my best to make this discussion self contained as far as possible so that the readers don’t have to look up to different references to follow the proofs. It is based on the presentation that I did in Measure Theory course in MStat 1st year.

Introduction

In statistics while modelling continuous data we assume that the data are iid observations coming from some ‘nice’ distribution. Then we do statistical inference. The strongest statistical argument we usually use is the Central Limit Theorem, which states that the sum of a large number of iid variables from a finite variance distribution will tend to be normally distributed. But this finite variance assumption is not always true. For example many real life data, such as financial assets exhibit fat tails. Such finite variance assumption does not hold for heavy tails. So there is a need for an alternative model. One such alternative is the Stable distribution. Of course there are other alternatives. But one good reason to use Stable distribution as an alternative is that they are supported by General Central Limit Theorem.

Notations: U\stackrel{d}{=} V means U and V have same distribution. Throughout this section we assume

\displaystyle X,X_1,X_2,\ldots \stackrel{iid}{\sim} F \ \ \ \mbox{and} \ S_n=\sum_{i=1}^n X_i

Motivation: Suppose F= \mbox{N}(0,\sigma^2). In that case we know

S_n\stackrel{d}{=} \sqrt{n}X

Motivated from the above relation we question ourselves that can we get a distribution such that the above relation holds with some other constants like n or \log n or n^{1/3} instead of \sqrt{n}? Hence we generalise the above relation in the following manner.

Definition: The distribution F is stable (in the broad sense) if for each n there exist constants c_n >0 and \gamma_n such that

S_n \stackrel{d}{=} c_nX+\gamma_n

and F is non degenerate. F is stable in the strict sense if the above relation holds with \gamma_n=0.

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