Tail Measures and Regular Variation

TL;DR

This paper develops a general framework for studying regular variation of measures and processes on Polish metric spaces, introducing tail measures and extending results to cadlag processes, with potential applications and open questions.

Contribution

It introduces a new approach to regular variation using tail measures on Polish spaces, extending existing results to multivariate cadlag processes and broadening the theoretical foundation.

Findings

Defined tail measures on spaces indexed by homogeneous coordinate maps

Extended regular variation results to multivariate cadlag processes

Provided insights into applications and open problems in the field

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.