Representation of the density functions of a multidimensional strictly stable distributions by series of generalized functions

S. V. Arhipov

Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 1 (2021), pp. 33-47

Résumé

The article discusses multidimensional strictly stable distributions. As is known, the density functions of these laws are not represented in closed form, with the exception of the well-known laws of Gauss and Cauchy. Characteristic functions are the starting point for research. There are several different forms of their presentation. The article chooses the form proposed in [1]. The application of the inverse Fourier transform together with the Abel summation of the integrals made it possible to obtain expansions of the density functions of multidimensional stable distributions (see [1], [12]). The main result of the article is the representation of these functions using series of generalized functions over the Lizorkin space. They make it possible to determine the order of decay of the principal term of the expansion at infinity for any radial direction. In addition, the derived formulas make it possible to see the structure of the formation of terms in expansions. In the corollary, examples are given for various cases of the support of the spectral measure of multidimensional stable laws.

Détail

Keywords: multidimensional strictly stable distributions, series of generalized functions, behavior of the density function for different radial directions.