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Spatial Cauchy processes with local tail dependence

We study a class of models for spatial data obtained using Cauchy convolution processes with random indicator kernel functions. We show that the resulting spatial processes have some appealing dependence properties including tail dependence at smaller distances and asymptotic independence at larger distances. We derive extreme-value limits of these processes and consider some interesting special cases. We show that estimation is feasible in high dimensions and the proposed class of models allows for a wide range of dependence structures.