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 Probability Surveys > Vol. 11 (2014) open journal systems 

Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps

Filip Lindskog, KTH Royal Institute of Technology, Stockholm
Sidney I. Resnick, Cornell University, School of Operations Research and Information Engineering
Joyjit Roy, Cornell University, School of Operations Research and Information Engineering

We develop a framework for regularly varying measures on complete separable metric spaces \(\mathbb{S}\) with a closed cone \(\mathbb{C}\) removed, extending material in [15, 24]. Our framework provides a flexible way to consider hidden regular variation and allows simultaneous regular-variation properties to exist at different scales and provides potential for more accurate estimation of probabilities of risk regions. We apply our framework to iid random variables in \(\mathbb{R}_+^\infty\)\(\) with marginal distributions having regularly varying tails and to càdlàg Lèvy processes whose Lèvy measures have regularly varying tails. In both cases, an infinite number of regular-variation properties coexist distinguished by different scaling functions and state spaces.

AMS 2000 subject classifications: 28A33,60G17,60G51,60G70

Keywords: Regular variation, multivariate heavy tails, hidden regular variation, tail estimation, L'evy process

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Lindskog, Filip, Resnick, Sidney I., Roy, Joyjit, Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps, Probability Surveys, 11, (2014), 270-314 (electronic). DOI: 10.1214/14-PS231.


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