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 S with a closed cone C removed, extending material in [14,23]. 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 R ∞ + 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.2010 Mathematics Subject Classification. 28A33,60G17,60G51,60G70.

Hidden regular variation under full and strong asymptotic dependence

“…The theory is flexible when given for closed subcones of metric spaces [23]; we specialize to subcones of R 2 + and R 2 where statistical results are most readily exhibited. Statistical extensions to higher dimensions are possible and require more sophisticated graphics.…”

Models with Hidden Regular Variation: Generation and Detection

“…The mathematical framework for the study of multivariate heavy tails is regular variation of measures. The theory is flexible when given for closed subcones of metric spaces [21], but we specialize to subcones of R 2 + where statistical results are most readily exhibited. Statistical extensions to higher dimensions will be discussed elsewhere.…”

“…The definitions of multivariate regular variation (MRV) and hidden regular variation (HRV) are reviewed in Section 1.4 where general concepts are adapted for subcones in two dimensions. In Section 2, assuming asymptotic limit measures are specified, we adapt the standard multiplicative method for generating regularly varying models based on the generalized polar coordinate transform [10,21,26] to E = R 2 + \ {0} and E 0 = (0, ∞) 2 producing relatively tractable models.…”

“…C(C \ C 0 ) Continuous, bounded, positive functions on C \ C 0 whose supports are bounded away from the forbidden zone C 0 . Without loss of generality [21], we may assume the functions are uniformly continuous. [10,18,21] and Definition 1.1.…”

“…We review material from [10,18,21] describing the framework for the definition of MRV and HRV specialized to two dimensions. Note that the convergence concept used for defining regular variation is M-convergence which is slightly different from vague convergence traditionally used in such cases.…”