Symmetric Stable Distribution
A symmetric stable distribution is defined through its
characteristic function as follows.
Definition 4.1: [Symmetric
stable distribution] A random variable X has a symmetric
stable distribution if and only if its characteristic function has the
form
Equation 4.165
where
Thus, a symmetric stable random variable is completely
characterized by three parameters, a, g, and q, where:
-
a is
called the characteristic exponent, which
indicates the "heaviness" of the tails of the distribution—a small value of
a implies a heavier
tail. The case a = 2
corresponds to a Gaussian distribution, whereas a = 1corresponds to a Cauchy distribution.
-
g is called the dispersion. For the Gaussian case (i.e., a = 2), g = ½ Var(X).
-
q is a location parameter, which is the mean when 1 < a 
2 and the median when 0 < a < 1.
By taking the Fourier transform of the characteristic function,
we can obtain the probability density function (pdf) of the symmetric stable
random variable X:
Equation 4.166
No closed-form expressions exist for general stable pdf's
except for the Gaussian (a = 2) and Cauchy (a = 1) pdf's. For these two pdf's, closed-form
expression exist:
Equation 4.167
Equation 4.168
It is known that for a non-Gaussian (a < 2) symmetric stable
random variable X with location parameter q = 0 and dispersion g, we have the
asymptote
Equation 4.169
where C(a) is a positive constant
depending on a. Thus,
stable distributions with a < 2 have inverse power tails, whereas Gaussian
distributions have exponential tails. Hence the tails of the stable
distributions are significantly heavier than those of the Gaussian
distributions. In fact, the smaller is a, the slower does its tail drop to zero, as shown in
Figs. 4.15 and 4.16.
As a consequence of (4.169), stable distributions do not have second-order
moments except for the limiting case of a = 2. More specifically, let X be a symmetric stable random variable with
characteristic exponent a. If 0 < a < 2, then
Equation 4.170
If a =
2, then
Equation 4.171
for all m 
0. Hence for 0 < a 1, stable distributions have no finite first- or higher-order moments;
for 1 < a < 2,
they have the first moments; and for a = 2, all moments exist. In particular, all
non-Gaussian stable distributions have in finite variance. The reader is
referred to [357]
for further details of these properties of a-stable distribution.
Generation of Symmetric Stable Random Variables
The following procedure generates a standard symmetric stable
random variable X with characteristic exponent
a, dispersion g = 1 and location parameter q = 0
(see [357]):
Equation 4.172
Equation 4.173
Equation 4.174
Equation 4.175
Equation 4.176
Equation 4.177
Equation 4.178
Equation 4.179
Now in order to generate a symmetric stable random variable
Y with parameters (a, g, q), we first generate a standard symmetric stable random
variable X with parameters (a, 1, 0), using the procedure
above. Then Y can be generated from X according to the following transformation:
Equation 4.180
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