Toward a Hazard Rate Framework for Regular and Rapid Variation
Haijun Li
TL;DR
This paper addresses how to characterize univariate tail variation in extreme value theory through a hazard-rate lens. By introducing generalized hazard rates $h(t)$ and their reciprocals $g(t)$, it unifies regular variation (Fréchet domain) and rapid variation (Gumbel domain) by linking tail decay to hazard-based conditions, including a Beurling-type scale $R(t)$ for rapid variation. It develops representations in the $RV_\rho$ and $\Gamma_\alpha(g)$ classes, with a hazard-rate–driven von Mises-type condition $\lim_{t\to\infty} \frac{g(t)f'(t)}{f(t)}=\alpha$ governing rapid variation and tail behavior. The framework extends to multivariate extremes via copula tail dependence and Beurling-type generalizations, offering a coherent, hazard-rate–centered toolkit for analyzing tail decay rates and their implications for domains of attraction and tail dependence in higher dimensions.
Abstract
Regular and rapid variation have been extensively studied in the literature and applied across various fields, particularly in extreme value theory. In this paper, we examine regular and rapid variation through the lens of generalized hazard rates, with a focus on the behavior of survival and density functions of random variables. Motivated by the von Mises condition, our hazard rate based framework offers a unified approach that spans from slow to rapid variation, providing in particular new insights into the relationship between hazard rate functions and the right tail decays of random variables.