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The calculation of the probability density and distribution function of a strictly stable law in the vicinity of zero
Viacheslav V. Saenko · 2022-10-13 · via math.PR updates on arXiv.org

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 $α<1$ the obtained series were asymptotic at $x\to0$, 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 $x\to0$ the obtained expansions can be successfully used to calculate the probability density and distribution function of strictly stable laws.

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