3.3 STABLE DISTRIBUTIONS AND SCALE
INVARIANCE
The principal property of stable distribution is that the sum of
variates has the same distribution shape as that of addends (see,
e.g., [6] for details). Both the Cauchy distribution and the normal
distribution are stable. This means, in particular, that the sum of
two normal distributions with the same mean and variance is also the
normal distribution (see Exercise 2). The general definition for
the stable distributions was given by Levy. Therefore, the stable
distributions are also called the Levy distributions.
Consider the Fourier transform F(q) of the probability distribution
function f(x)
F(q) ¼
ð
f(x)e
iqx
dx (3:3:1)
The function F(q) is also called the characteristic function of the
stochastic process. It can be shown that the logarithm of the charac-
teristic function for the Levy distribution has the following form
ln F
L
(q) ¼
imq gjqj
a
[1 ibd tan (pa=2)], if a 1
imq gjqj[1 þ 2ibd ln (jqj)=p)], if a ¼ 1
(
(3:3:2)
In (3.3.2), d ¼ q=jqj and the distribution parameters must satisfy the
following conditions
0 < a 2, 1 b 1, g > 0(3:3:3)
The parameter m corresponds to the mean of the stable distribution
and can be any real number. The parameter a characterizes the
distribution peakedness. If a ¼ 2, the distribution is normal. The
parameter b characterizes skewness of the distribution. Note that
skewness of the normal distribution equals zero and the parameter
b does not affect the characteristic function with a ¼ 2. For the
normal distribution
ln F
N
(q) ¼ imq gq
2
(3:3:4)
The non-negative parameter g is the scale factor that characterizes the
spread of the distribution. In the case of the normal distribution,
g ¼ s
2
=2 (where s
2
is variance). The Cauchy distribution is defined
Probability Distributions 25
with the parameters a ¼ 1 and b ¼ 0. Its characteristic function
equals
ln F
C
(q) ¼ imq gjqj (3:3:5)
The important feature of the stable distributions with a < 2 is that
they exhibit the power-law decay at large absolute values of the
argument x
f
L
(jxj) jxj
(1þa)
(3:3:6)
The distributions with the power-law asymptotes are also named the
Pareto distributions. Many processes exhibit power-law asymptotic
behavior. Hence, there has been persistent interest to the stable distri-
butions.
The power-law distributions describe the scale-free processes. Scale
invariance of a distribution means that it has a similar shape on
different scales of independent variables. Namely, function f(x) is
scale-invariant to transformation x ! ax if there is such parameter
L that
f(x) ¼ Lf(ax) (3:3:7)
The solution to equation (3.3.7) is simply the power law
f(x) ¼ x
n
(3:3:8)
where n ¼ln (L)= ln (a). The power-law function f(x) (3.3.8) is scale-
free since the ratio f(ax)=f(x) ¼ L does not depend on x. Note that the
parameter a is closely related to the fractal dimension of the function
f(x). The fractal theory will be discussed in Chapter 6.
Unfortunately, the moments of stable processes E[x
n
] with power-
law asymptotes (i.e., when a < 2) diverge for n a. As a result, the
mean of a stable process is infinite when a 1. In addition, variance
of a stable process is infinite when a < 2. Therefore, the normal
distribution is the only stable distribution with finite mean and finite
variance.
The stable distributions have very helpful features for data analysis
such as flexible description of peakedness and skewness. However, as it
was mentioned previously, the usage of the stable distributions in
financial applications is often restricted because of their infinite vari-
ance at a < 2. The compromise that retains flexibility of the Levy
26 Probability Distributions

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