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., Mantegna & Stanley 2000; Bouchaud & Potters 2000). 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. The general definition for the stable distributions was given by Levy (hence, another name—the Levy distribution). Consider the Fourier transform F (q) of the probability distribution function f(x):

The function F(q) is also called the characteristic function of stochastic process. It can be shown that the logarithm of the characteristic function for the Levy distribution has the following form: