A general framework for the study of regular variation (RV) is that of Polish star-shaped metric spaces, while recent developments in  have discussed RV with respect to a properly localised boundedness . Along the lines of the latter approach, we discuss the RV of Borel measures and random processes on a general Polish metric spaces . Tail measures introduced in  appear naturally as limiting measures of regularly varying time series. We define tail measures on the measurable space indexed by , a countable family of 1-homogeneous coordinate maps, and show some tractable instances for the investigation of RV when is determined by . This allows us to study the regular variation of càdlàg processes on retrieving in particular results obtained in  for RV of stationary càdlàg processes on the real line removing therein. Further, we discuss potential applications and open questions.
MB kindly acknowledges support from the SNSF Grant 200021-191984 and EH, GS kindly acknowledge support from the SNSF Grant 1200021-196888.
We are grateful to both reviewers and the Editor for several comments and suggestions that lead to significant improvement of the manuscript. Many thanks go to Richard Davis, Philippe Soulier, Thomas Mikosch and Ilya Molchanov for literature suggestions and discussions.
"Tail measures and regular variation." Electron. J. Probab. 27 1 - 43, 2022. https://doi.org/10.1214/22-EJP788