ABSTRACT
Abstract Simulation of stochastic phenomena has become an important tool to assess characteristics which are analytically intractable. In this chapter, we deal with the problem of sampling from max-stable processes, which, by the spectral representation (de Haan, 1984), require taking the pointwise maxima over an infinite number of spectral functions belonging to a Poisson point process. In practice, spectral functions are simulated in an appropriate order until some stopping rule takes effect. For mixed moving maxima processes with bounded shape functions with joint compact support, Schlather (2002) provides such a stopping criterion yielding an exact simulation. Oesting et al. (2013) consider stopping rules for a family of equivalent spectral representations in a very general setting, particularly focusing on the normalized spectral representation which allows for an exact simulation, as well. Although this representation exists under mild conditions, the distribution of the corresponding spectral functions might be inappropriate for sampling. In this case, approximative procedures are proposed. Here, the choice of the stopping rule and the spectral representation is crucial for the quality of approximation. In this context, we discuss measures of simulation efficiency and quality. Several examples of max-stable processes are analyzed. We particularly focus on the challenging case of Brown-Resnick processes which are stationary although originally constructed via nonstationary spectral functions. Finally, we review existing R packages on the simulation of max-stable processes.
