Tail Measures And Regular Variation

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 withrespect to a properly localised boundednessB. Along the lines of the latter approach,we discuss the RV of Borel measures and random processes on a general Polish metricspaces(D, dD). Tail measures introduced in [47] appear naturally as limiting measuresof regularly varying time series. We define tail measures on the measurable space(D,D)indexed byH(D), a countable family of 1-homogeneous coordinate maps, andshow some tractable instances for the investigation of RV whenBis determined byH(D). This allows us to study the regular variation of càdlàg processes onD(Rl,Rd)retrieving in particular results obtained in [59] for RV of stationary càdlàg processeson the real line removingl= 1therein. Further, we discuss potential applications andopen questions.