The index of stability is the shape parameter 
, with 
, of a nondegenerate stable
distribution. It is also called the stability index or characteristic
exponent. If 
,
..., 
are statistically independent copies of
a random variable 
with this stable distribution,
then
 |
where 
is a constant and equality denotes equality of statistical
distributions (Nolan 2020).
The case 
gives a normal distribution, while 
with skewness parameter 
gives a Cauchy
distribution. For 
, at least one tail
probability decays as a positive constant times 
as 
, and the variance is
infinite (Nolan 2020).
