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

TL;DR

This paper derives power series expansions for the probability density and distribution functions of strictly stable laws near zero, revealing different convergence behaviors depending on the stability parameter.

Contribution

It provides explicit power series expansions for stable laws near zero, clarifying their convergence properties for different alpha values.

Findings

Series are asymptotic for α<1 as x→0

Series are convergent for α>1 as x→0

Series converge to asymmetric Cauchy distribution for α=1

Abstract

The problem of calculating the probability density and distribution function of a strictly stable law is considered at $x\to0$. The expansions of these values into power series were obtained to solve this problem. It was shown that in the case $\alpha<1$ the obtained series were asymptotic at $x\to0$, in the case $\alpha>1$ they were convergent and in the case $\alpha=1$ in the domain $|x|<1$ these series converged to an asymmetric Cauchy distribution. It has been shown that at $x\to0$ the obtained expansions can be successfully used to calculate the probability density and distribution function of strictly stable laws.