Tail Measures and Regular Variation

DOI10.1214/22-EJP788zbMATH Open1514.60063arXiv2103.04396MaRDI QIDQ6362336

Martin Bladt, Georgiy Shevchenko, Enkelejd Hashorva

Publication date: 7 March 2021

Abstract: A general framework for the study of regular variation (RV) is that of Polish star-shaped metric spaces, while recent developments in [1] have discussed RV with respect to some properly localised boundedness $mathcalB$ imposing weak assumptions on the structure of Polish space. Along the lines of the latter approach, we discuss the RV of Borel measures and random processes on general Polish metric spaces. Tail measures introduced in [2] appear naturally as limiting measures of regularly varying time series. We define tail measures on a measurable space indexed by $mathcalH(D)$, a countable family of homogeneous coordinate maps, and show some tractable instances for the investigation of RV when $mathcalB$ is determined by $mathcalH(D)$. This allows us to study the regular variation of cadlag processes on $D(R^l,R^d)$ retrieving in particular results obtained in [1] for RV of stationary cadlag processes on the real line removing $l=1$ therein. Further, we discuss potential applications and open questions.

Mathematics Subject Classification ID

Central limit and other weak theorems (60F05) Extreme value theory; extremal stochastic processes (60G70) Spaces of measures, convergence of measures (28A33) Convergence of probability measures (60B10)

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