2021/01/01 by Günter Last, Last, Günter · 1 citation
Economics, Econometrics and Finance · Mathematics · #60G57 #60G70 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2112.15380
openalex publication_date 2021/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using an intrinsic approach, we study some properties of random fields which appear as tail fields of regularly varying stationary random fields. The index set is allowed to be a general locally compact Hausdorff Abelian group \mathbbG. The values are taken in a measurable cone, equipped with a pseudo norm. We first discuss some Palm formulas for the exceedance random measure ξ associated with a stationary (measurable) random field Y=(Ys)s∈G. It is important to allow the underlying stationary measure to be σ-finite. Then we proceed to a random field (defined on a probability space) which is spectrally decomposable, in a sense which is motivated by extreme value theory. We characterize mass-stationarity of the exceedance random measure in terms of a suitable version of the classical Mecke equation. We also show that the associated stationary measure is homogeneous, that is a tail measure. We then proceed with establishing and studying the spectral representation of stationary tail measures and with characterizing a moving shift representation. Finally we discuss anchoring maps and the candidate extremal index.