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  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.
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
, which are independent and identically distributed according to
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:
, , 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
and has finite absolute moments of order
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
, its support lies in the intervals
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.