Tail Measures and Regular Variation

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 $\mathcal{B}$ 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 $\mathcal{H}(D)$, a countable family of homogeneous coordinate maps, and show some tractable instances for the investigation of RV when $\mathcal{B}$ is determined by $\mathcal{H}(D)$. 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.

arXiv comment37 pages, published

References in corpus (12)

• Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes
• Ergodic properties of sum- and max-stable stationary random fields via null and positive group actions
• Invariance properties of random vectors and stochastic processes based on the zonoid concept
• A note on vague convergence of measures
• The tail process and tail measure of continuous time regularly varying stochastic processes
• Approximation of Supremum of Max-Stable Stationary Processes and Pickands Constants
• Group theoretic dimension of stationary symmetric α-stable random fields
• On Extremal Index of Max-Stable Random Fields
• Polar decomposition of scale-homogeneous measures with application to Lévy measures of strictly stable laws
• Tail processes for stable-regenerative multiple-stable model
• Weak Convergence of Probability Measures
• Estimation of cluster functionals for regularly varying time series: runs estimators

Cited by in corpus (2)

• Cluster Random Fields and Random-Shift Representations
• Shift-invariant homogeneous classes of random fields