## Probability Surveys

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

#### Abstract

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 Lévy processes whose Lévy measures have regularly varying tails. In both cases, an infinite number of regular-variation properties coexist distinguished by different scaling functions and state spaces.

#### Article information

Source
Probab. Surveys, Volume 11 (2014), 270-314.

Dates
First available in Project Euclid: 21 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.ps/1413896892

Digital Object Identifier
doi:10.1214/14-PS231

Mathematical Reviews number (MathSciNet)
MR3271332

Zentralblatt MATH identifier
1317.60007

#### Citation

Lindskog, Filip; Resnick, Sidney I.; Roy, Joyjit. Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps. Probab. Surveys 11 (2014), 270--314. doi:10.1214/14-PS231. https://projecteuclid.org/euclid.ps/1413896892

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