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 [41] have discussed RV with respect to a properly localised boundedness $\mathcal{B}$. Along the lines of the latter approach, we discuss the RV of Borel measures and random processes on a general Polish metric spaces $(\mathrm{D},d_{\mathrm{D}})$. Tail measures introduced in [47] appear naturally as limiting measures of regularly varying time series. We define tail measures on the measurable space $(\mathrm{D},\mathcal{D})$ indexed by $\mathcal{H}(\mathrm{D})$, a countable family of 1-homogeneous coordinate maps, and show some tractable instances for the investigation of RV when $\mathcal{B}$ is determined by $\mathcal{H}(\mathrm{D})$. This allows us to study the regular variation of càdlàg processes on $D({\mathbb{R}}^l,{\mathbb{R}}^d)$ retrieving in particular results obtained in [59] for RV of stationary càdlàg processes on the real line removing $l=1$ therein. Further, we discuss potential applications and open questions.