Extremal behaviour of aggregated data with an application to downscaling

SUMMARY 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 $\ell$ we introduce the $\ell$-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 $\ell$-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.