The paper considers the problem of calculating the distribution function of a strictly stable law at $x\to\infty$. To solve this problem, an expansion of the distribution function in a power series was obtained, and an estimate of the remainder term was also obtained. It was shown that in the case $\alpha<1$ this series was convergent for any $x$, in the case $\alpha=1$ the series was convergent at $N\to\infty$ in the domain $|x|>1$, and in the case $\alpha>1$ the series was asymptotic at $x\to\infty$. The case $\alpha=1$ was considered separately and it was demonstrated that in that case the series converges to the generalized Cauchy distribution. An estimate for the threshold coordinate $x_\varepsilon^N$ was obtained which determined the area of applicability of the obtained expansion. It was shown that in the domain $|x|\geqslant x_\varepsilon^N$ this power series could be used to calculate the distribution function, which completely solved the problem of calculating the distribution function at large $x$.
翻译：本文考虑了计算严格稳定概率分布在$x\to\infty$下的分布函数问题。为解决此问题，我们得到了分布函数在幂级数中的展开式，并获得了余项的估计。结果表明，在$\alpha<1$的情况下，该级数对任何$x$都是收敛的，在$\alpha=1$的情况下，该级数在$|x|>1$的域中$N\to\infty$收敛，在$\alpha>1$的情况下，该级数在$x\to\infty$时呈现出渐近性。$\alpha=1$的情况被单独考虑，并展示了在该情况下，级数收敛到广义柯西分布。我们得到了阈值坐标$x_\varepsilon^N$的估计，用以确定所获得展开式的适用区域。结果表明，在$|x|\geqslant x_\varepsilon^N$的域中，可以使用该幂级数来计算分布函数，从而完全解决了在大$x$处计算分布函数的问题。
相关内容
微信扫码咨询专知VIP会员