The tail process and tail measure of continuous time regularly varying\n stochastic processes

2020/04/01 by Philippe Soulier, Soulier, Philippe · 1 citation
Economics, Econometrics and Finance · Mathematics · #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.2004.00325

openalex publication_date 2020/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The goal of this paper is to investigate the tools of extreme value theory\noriginally introduced for discrete time stationary stochastic processes (time\nseries), namely the tail process and the tail measure, in the framework of\ncontinuous time stochastic processes with paths in the space \D of\nc `adl `ag functions indexed by \ℝ, endowed with Skorohod's J₁\ntopology. We prove that the essential properties of these objects are\npreserved, with some minor (though interesting) differences arising. We first\nobtain structural results which provide representation for homogeneous\nshift-invariant measures on \D and then study regular variation of\nrandom elements in \D. We give practical conditions and study\nseveral examples, recovering and extending known results.\n

Citations

• Regularly varying time series in Banach spaces
• Ergodic decompositions of stationary max-stable processes in terms of\n their spectral functions

Cited by

• Shift-generated α-homogeneous classes of jointly measurable random fields

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