ABSTRACT

This article investigates general scaling settings and limit distributions of functionals of filtered random fields. The filters are defined by the convolution of non-random kernels with functions of Gaussian random fields. The case of long-range dependent fields and increasing observation windows is studied. The obtained limit random processes are non-Gaussian. Most known results on this topic give asymptotic processes that always exhibit non-negative auto-correlation structures and have the self-similar parameter $H\in (\frac{1}{2},1)$. In this work, we also obtain convergence for the case $H\in (0,\frac{1}{2})$ and show how the Hurst parameter H can depend on the shape of the observation windows. Various examples are presented.

摘要

本文研究了常规缩放设置和已过滤随机字段功能的极限分布。滤波器是由非随机核与高斯随机场函数的卷积定义的。研究了远程依赖场和观察窗增加的情况。所获得的极限随机过程是非高斯的。关于该主题的最著名结果给出了渐近过程，这些过程总是表现出非负自相关结构并具有自相似参数$H\in （\frac{\mathrm{1个}}{2}，\mathrm{1个}）$。在这项工作中，我们还获得了案例的收敛性$H\in （0，\frac{\mathrm{1个}}{2}）$并展示赫斯特参数H如何取决于观察窗的形状。给出了各种示例。