Dispersion measure for symmetric, stable probability distributions

Abstract

Stable distributions is an interesting and important class of probability distributions. They were discovered explicitly by Paul Lévy in 1925 \cite{lk}. They possess many interesting properties, most importantly they are by definiton invariant under addition, up to a scale. Noteworthly they have power-law type of decay and therefore they are an excellent model for modelling many natural phenomena, such as earthquakes, financial returns, and a multitude of social phenomena such as size distributions of cities and firms \cite{scaling}. The major problem concerning them is that they have an infinite variance \cite{GK} and therefore their practical applicability is somewhat limited. Also they generally do not possess a density expressible in an analytic form. This study proposes a dispersion measure for them, drawing ideas from Fisher information, differential geometry and most importantly, the uncertainty principle for Fourier transform pairs \cite{Weyl}. The study begins with a brief discussion on characteristic functions and their relation to Fourier transforms and their properties, proceeds to a brief presentation of stable distributions and accumulates in defining a concept of \textit{characteristic curvature}, which is proposed as a suitable measure of dispersion for class of stable distributions. Characteristic curvature satisfies the familiar scaling property for stable processes, reduces to that of standard deviation in the Gaussian case and is explicitly related to the Euler Gamma function, the Hurst exponent H and the fractal dimension 2-H.