Extremal behaviour of aggregated data with an application to downscaling

The distribution of spatially aggregated data from a stochastic process |X| may exhibit tail behaviour different from that of its marginal distributions. For a large class of aggregating functionals || we introduce the ||-extremal coefficient, which quantifies this difference as a function of the extremal spatial dependence in |X|⁠. We also obtain the joint extremal dependence for multiple aggregation functionals applied to the same process. Formulae for the ||-extremal coefficients and multivariate dependence structures are derived in important special cases. The results provide a theoretical link between the extremal distribution of the aggregated data and the corresponding underlying process, which we exploit to develop a method for statistical downscaling. We apply our framework to downscale daily temperature maxima in the south of France from a gridded dataset and use our model to generate high-resolution maps of the warmest day during the |$2003$| heatwave.