论文标题
空间随机场的极端聚类和聚类计数
Extremal clustering and cluster counting for spatial random fields
论文作者
论文摘要
我们考虑了一个固定的随机场，该字段通过$ \ mathbb {z}^d $的增加的子集的序列索引，对序列的扩展方式遵守非常广泛的几何假设。在某些混合和局部条件下，我们展示了各个变量的尾巴分布如何与索引集在索引集扩展的索引集上的最大磁场的尾部行为相关联。 此外，在一个框架中，我们让增加的索引集为固定集合$ c $的标量乘法，可能在不同方向上具有不同的标量，我们使用两个群集定义来定义重新定义的群集计数点进程$ c $；一个群集定义将设置的索引划分为越来越多的框，如果包含极端观察结果，则将框架算作盒子。另一个更直观的集群定义认为，如果距离接近，则将极端点在同一集群中。我们表明，两个群集点过程都将$ c $上的泊松点流程收敛到泊松点。此外，我们找到了平均簇大小的限制。最后，我们在没有集群的情况下特别注意此案。
We consider a stationary random field indexed by an increasing sequence of subsets of $\mathbb{Z}^d$ obeying a very broad geometrical assumption on how the sequence expands. Under certain mixing and local conditions, we show how the tail distribution of the individual variables relates to the tail behavior of the maximum of the field over the index sets in the limit as the index sets expand. Furthermore, in a framework where we let the increasing index sets be scalar multiplications of a fixed set $C$, potentially with different scalars in different directions, we use two cluster definitions to define associated cluster counting point processes on the rescaled index set $C$; one cluster definition divides the index set into more and more boxes and counts a box as a cluster if it contains an extremal observation. The other cluster definition that is more intuitive considers extremal points to be in the same cluster, if they are close in distance. We show that both cluster point processes converge to a Poisson point process on $C$. Additionally, we find a limit of the mean cluster size. Finally, we pay special attention to the case without clusters.
