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 $x\to 0$. 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 $x\to 0$, 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 $x\to 0$ the obtained expansions can be successfully used to calculate the probability density and distribution function of strictly stable laws.

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