Extremal behavior of stationary Gaussian sequences/random fields is widely investigated since it models common cluster phenomena and brings a bridge between discrete and continuous extremes. We establish extensively limit theorems of stationary random fields under certain mixing and dependence conditions, which are further illustrated by typical examples of order statistics of Gaussian random fields and skew-Gaussian random fields. The positivity of the cluster index involved and its link with the expected cluster size are discussed.

中文翻译：

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晶格上的平稳随机场的极值

静态高斯序列/随机场的极值行为得到了广泛研究，因为它模拟了常见的簇现象并在离散和连续的极端之间建立了桥梁。我们建立了在一定混合和相关条件下平稳随机场的广义极限定理，进一步通过高斯随机场和偏高斯随机场阶数统计的典型例子进一步说明。讨论了所涉及的聚簇索引的正性及其与预期聚簇大小的联系。