Tail processes and tail measures: An approach via Palm calculus

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 G. We first discuss some Palm formulas for the exceedance random measure ξ associated with a stationary (measurable) real-valued 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 proving that any stationary tail measure has a spectral representation and with characterizing a moving shift representation. Finally we discuss the special case of a discrete group. All results can be extended to random fields with values in general measurable cones.

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