We study an independence test based on distance correlation for random fields $(X,Y)$. We consider the situations when $(X,Y)$ is observed on a lattice with equidistant grid sizes and when $(X,Y)$ is observed at random locations. We provide \asy\ theory for the sample distance correlation in both situations and show bootstrap consistency. The latter fact allows one to build a test for independence of $X$ and $Y$ based on the considered discretizations of these fields. We illustrate the performance of the bootstrap test in a simulation study involving fractional Brownian and infinite variance stable fields. The independence test is applied to Japanese meteorological data, which are observed over the entire area of Japan.

翻译：我们根据随机字段(X,Y)的距离相关关系研究独立测试。 我们考虑的是,当(X,Y)美元在等距离网格大小的平面上观察到时,当(X,Y)美元在随机地点观察到时,我们为两种情况下的抽样距离相关关系提供asy\理论,并显示靴带的一致性。后一事实允许人们根据考虑的这些字段的离散性,建立X美元和Y美元的独立测试。我们在涉及分数布朗和无限差异稳定场的模拟研究中展示了靴带测试的性能。独立测试适用于日本气象数据,日本气象数据在日本整个地区都观测过。