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Appendix A L´evy Distributions


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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.


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.



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 



 ) so that the distribution of the

nor-malised sum,                (A.3)

converges to a limiting distribution  


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


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,



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


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 


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


 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



In particular, we have       and      for a Gaussian.

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


) . 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


  which have the same asymptotic behaviour for large

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


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