Stable Distributions

3. Other features

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3.1          Three special cases of stable laws, which have closed form expressions for their probability densities are:


(a)    Normal (i.e. Gaussian).  if  has density



(b)   Cauchy.  if  has density



(c)    Levy.  if  has density



3.2          Generic features of stable distributions noted by Nolan (2005) include:


(a)    They are unimodal


(b)   When  is small then the skewness parameter is significant, but when  is close to 2 then it matters less and less.


(c)    When  (i.e. the Normal distribution), the distribution has ‘light’ tails and all moments exist. In all other cases (i.e. ), stable distributions have heavy tails and an asymptotic power law (i.e. Pareto) decay. The term stable Paretian is thus used to distinguish the  case from the Normal case. A consequence of these heavy tails is that not all population moments exist. If  then the population variance does not exist, and if  then the population mean does not exist either. Fractional moments, e.g. the ’th absolute moment, defined as , exist if and only if  (if ). Of course all sample moments exist, if there are sufficient observations in the sample, but these may exhibit unstable behaviour as the sample size increases if the corresponding population moment does not exist.


3.3          Linear combinations of independent stable distributions with the same index, , are stable. If  for  then






In this generalisation of the definition of stable distributions given in Section 1.3 it is essential for the ’s to be the same. Adding stable random variables with different ’s does not result in a stable law.


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