Tail Measures and Regular Variation

Martin Bladt, Enkelejd Hashorva, Georgiy Shevchenko

Published 2021-03-07Version 1

A general framework for the study of regular variation is that of Polish star-shaped metric spaces, while recent developments in [1] have discussed regular variation in relation to a boundedness and weaker assumptions are imposed therein on the structure of Polish space. Along the lines of the latter approach, we discuss the regular variation of measures and processes on Polish spaces with respect to some given boundedness. We then focus on regular variation of cadlag processes on D(R^l, \R^d), which was recently studied for stationary cadlag processes on the real line in [2]. Tail measures introduced in [3] appear naturally as limiting measures of regularly varying random processes, and we show that this continues to hold true in our general setting. We derive several results for tail measures and their local tail/ spectral tail processes which are crucial for the investigation of regular variation of Borel measures.