Moving average processes driven by exponential-tailed L\'evy noise are important extensions of their Gaussian counterparts in order to capture deviations from Gaussianity, more flexible dependence structures, and sample paths with jumps. Popular examples include non-Gaussian Ornstein--Uhlenbeck processes and type G Mat\'ern stochastic partial differential equation random fields. This paper is concerned with the open problem of determining their extremal dependence structure. We leverage the fact that such processes admit approximations on grids or triangulations that are used in practice for efficient simulations and inference. These approximations can be expressed as special cases of a class of linear transformations of independent, exponential-tailed random variables, that bridge asymptotic dependence and independence in a novel, tractable way. This result is of independent interest since models that can capture both extremal dependence regimes are scarce and the construction of such flexible models is an active area of research. This new fundamental result allows us to show that the integral approximation of general moving average processes with exponential-tailed L\'evy noise is asymptotically independent when the mesh is fine enough. Under mild assumptions on the kernel function we also derive the limiting residual tail dependence function. For the popular exponential-tailed Ornstein--Uhlenbeck process we prove that it is asymptotically independent, but with a different residual tail dependence function than its Gaussian counterpart. Our results are illustrated through simulation studies.

翻译：指数尾部Lévy噪声驱动的移动平均过程是高斯过程的重要扩展，用于捕捉偏离高斯性的特征、更灵活的相依结构以及带有跳跃的样本路径。常见例子包括非高斯Ornstein-Uhlenbeck过程与G型Matérn随机偏微分方程随机场。本文关注确定其极值相依结构这一开放性问题。我们利用此类过程在用于高效模拟与推断的网格或三角剖分上存在近似这一特性。这些近似可表示为独立指数尾部随机变量线性变换类别的特殊情形，以一种新颖且易处理的方式桥接了渐近相依性与渐近独立性。该结果本身具有独立研究价值，因为能同时捕捉两种极值相依模式的模型十分稀缺，而构建此类灵活模型是当前活跃的研究领域。这一新基础性结果使我们能够证明：当网格足够精细时，带有指数尾部Lévy噪声的一般移动平均过程的积分近似具有渐近独立性。在核函数的温和假设下，我们还导出了极限残差尾部相依函数。对于经典的指数尾部Ornstein-Uhlenbeck过程，我们证明其具有渐近独立性，但其残差尾部相依函数与高斯版本不同。通过模拟研究验证了我们的结论。