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Abstract
We consider a class of stationary processes that exhibit both long-range dependence and heavy tails. While separate limit theorems for the partial sums and the maxima of such processes have recently been established—featuring novel limiting objects—this work develops the joint sum-and-max limit theorems for this class. In the finite-variance case, the limiting behavior consists of two independent components: a fractional Brownian motion (from the sum) and a long-range dependent random sup measure (from the maximum). In contrast, in the infinite-variance regime, the limit comprises two dependent components: a stable Lévy process and a random sup measure. Their dependence is characterized through the local time and range of a stable subordinator. To establish this result, we also prove a joint convergence theorem for the local time and range of subordinators, which may be of independent interest. In parallel, we investigate the estimation of multivariate extreme value models with a discrete spectral measure using spherical clustering techniques. The primary methodological contribution is a new order selection criterion—selecting the number of spectral atoms (or clusters)—based on an augmented silhouette width index. This criterion introduces a penalty term that discourages overly small clusters and insufficient separation between cluster centers. We prove that the method consistently recovers the true number of atoms in the spectral measure, enabling consistent estimation of the order of max-linear factor models, which lack standard likelihood-based tools for model selection. Our second contribution is a large deviation analysis that quantifies the convergence quality of clustering-based estimation of spectral measures. Finally, we demonstrate how the discrete spectral measure estimation can be translated into parameter estimation for heavy-tailed factor models, supported by simulations and real-world data examples that illustrate both order selection and model inference.