The calculation of the probability density and distribution function of a strictly stable law in the vicinity of zero

10/13/2022

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by   Viacheslav V. Saenko, et al.

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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.