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 6¼ 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 ¼ln (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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