The Calculation of the Probability Density and Distribution Function of a Strictly Stable Law in the Vicinity of Zero

Abstract

The problem of calculating the probability density and distribution function of a strictly stable law is considered at x→0. The expansions of these values into power series were obtained to solve this problem. It was shown that in the case α<1, the obtained series were asymptotic at x→0; in the case α>1, they were convergent; and in the case α=1 in the domain |x|<1, these series converged to an asymmetric Cauchy distribution. It has been shown that at x→0 the obtained expansions can be successfully used to calculate the probability density and distribution function of strictly stable laws.