We investigate long and short memory in $\alpha$-stable moving averages and max-stable processes with $\alpha$-Fr\'echet marginal distributions. As these processes are heavy-tailed, we rely on the notion of long range dependence suggested by Kulik and Spodarev (2019) based on the covariance of excursions. Sufficient conditions for the long and short range dependence of $\alpha$-stable moving averages are proven in terms of integrability of the corresponding kernel functions. For max-stable processes, the extremal coefficient function is used to state a necessary and sufficient condition for long range dependence.

中文翻译：

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稳定随机过程的长程相关性

我们研究了 $\alpha$-stable 移动平均线和具有 $\alpha$-Fr\'echet 边际分布的最大稳定过程中的长短记忆。由于这些过程是重尾的，我们依赖于 Kulik 和 Spodarev（2019）基于偏移协方差提出的长期依赖概念。在相应核函数的可积性方面证明了 $\alpha$ 稳定移动平均线的长期和短期相关性的充分条件。对于最大稳定过程，极值系数函数用于说明长程相关性的充分必要条件。