Spatial Deformation Models for Non-Stationary Max-Stable Processes

Max-stable processes constitute the natural class for modeling spatial extremes, but existing models almost all assume spatial stationarity, an assumption rarely verified in practice. This paper proposes a generalization of max-stable processes to the non-stationary framework using the spatial deformation technique. We develop a rigorous theoretical framework based on a generalization of de Haan’s spectral representation, enabling the construction of non-stationary max-stable processes from deformed stationary processes. We prove a fundamental characterization theorem: a max-stable process is non-stationary by deformation if and only if there exists a bijective transformation rendering it stationary, and we establish identifiability properties guaranteeing that the deformation is identifiable up to an isometry. Three classes of parametric deformations (piecewise affine, radial, and spline-based) are proposed, and we study the asymptotic properties of the extremal coefficient as well as consistent estimation methods. This work, complementary to the recent algorithmic approach of Richards and Wadsworth (2021), establishes the necessary mathematical foundations for modeling nonstationary extremes in climatology, hydrology, and epidemiology.