Appendix A L´evy Distributions

L ´evy Distributions

This appendix is meant to explain what is a L´evy or stable distribution. We wanted to give an introduction to the mathematics and properties of these distribution in order to lay stress on the importance that they have in many real data analysis.

A.1

Beyond the Central Limit Theorem

The Central Limit Theorem [7] states that a sum of

independent and identically distributed random variables



, with finite first and second moments, obeys a Gaussian distribution in the limit  . That is, if        

is the partial sum of the above random variables, the central limit theorem holds and



is distributed as a Gaussian in the limit

 .

Many distributions belong to the domain of attraction of the Gaussian, but not all. There is a whole class of distributions which not fulfils the hypothesis of a finite second moment such as:                  (A.1) where
 

) . This kind of inverse power law tails precludes the convergence to the

Gaussian distribution but not the existence of a limiting distribution.

A.2

Stable distributions

L´evy stable distributions arise from the generalisation of the central limit theorem to a wider class of distribution, see [7, 6].

Consider a set of random variables 



 , with





, which are independent and identically distributed according to

Prob

  
     


   (A.2)

It is possible to find (real) constants

 and

 (



) so that the distribution of the

nor-malised sum,   

      

(A.3)

converges to a limiting distribution  



 if



tends to infinity. Before showing this, it is required the definition of a stable distribution.

Definition A.1. A probability density is called ‘stable’ if it is invariant under convolution,

i.e., if there are constants



and such that

                                      
       (A.4)

for all (real) constants




  

  .

Foe example a Gaussian distribution satisfies this definition and therefore is stable. The importance of this characteristic is due to the following theorem by L´evy and Khintchine:

Theorem A.1. A probability density 



 can only be a limiting distribution of the sum A.3

of independent and randomly distributed random variables if it is stable.

The Gaussian distribution is therefore a potential limiting distribution. However there are many more. L´evy and Khintchine have completely specified the form of all possible stable distributions:

Theorem A.2 (Canonical representation). A probability density 







 is stable if the

logarithm of its characteristic function,

    
 
 
           

(A.5)

reads


   
           
  (A.6) where

, and  are real constants taking the values:

arbitrary, 
,



 ,  

 , and the function
  is given by

  
   

  for   


   for   (A.7) The constants

and are scale factors. Replacing

 
with  

shifts the origin and rescale the abscissa, but do not alter the function


  (unless   ,  
). In contrast,

and define the shape and the properties of





 . These parameters are therefore used

as indices to distinguish different stable distributions. The parameter characterises the large- behaviour of



 and determines which moments exists:

 

 : each stable distribution, also called L´evy distribution, behaves as


         for    (A.8)

and has finite absolute moments of order 

              

  if
    (A.9)

In particular, the latter property implies that the variance does not exist if 

 and

that both mean value and variance do not exist if

  .   :



 is independent of  , since
   
, and is Gaussian.

Due to these properties, is called characteristic exponent. The second characteristic pa-rameter, , determines the asymmetry of


  :  
:

  is an even function of  .     :



 exhibits a pronounced asymmetry for some choice of . For

in-stance, if

 

 , its support lies in the intervals


 for   and 

 

Theorem A.2 defines the general expression for all possible stable distributions. However, it does not specify the conditions which the probability density

  has to satisfy so that the

distribution of the normalised sum

  converges to a particular
 
 in the limit   .

If this is the case, one can say:

  belongs to the domain of attraction of





 .

Theorem A.1. The probability density

  belongs to the domain of attraction of a stable

density





 with characteristic exponent 
  )  if and only if           for  $  (A.10) where  
,  
and  
are constants.

These constants are directly related to the prefactor and the asymmetry parameter by


                  for           for   (A.11)  
         for             for    (A.12) Furthermore, if

  belongs to the domain of attraction of a stable distribution, its absolute

moment of order exist for

 :                 for

  
)   for     )   (A.13)

and the normalisation constant in A.3, which characterises the typical (scaling) behaviour of

 is given by      (A.14) so that      Prob 
     
 
 
 
 
 
 (A.15)

where is the same constant as in A.10 and

  
 
 

(A.16)

In particular, we have       and
      for a Gaussian.

Equations A.10 and A.13 echo the behaviour of the limiting distribution

    for 

) . This reflects the difference between the Gaussian and other stable distributions.

Whereas all probability densities which decay rapidly enough at large (at least as  





) belong to the attraction domain of the Gaussian, stable distribution with



) only attract

those

  which have the same asymptotic behaviour for large



. This restricting condition is a possible reason for the prevalence of the Gaussian in nature.