In this paper, we consider a financial or insurance system with a finite number of individual risks described by real-valued random variables. We focus on two kinds of risk measures, referred to as the tail moment (TM) and the tail central moment (TCM), which are defined as the conditional moment and conditional central moment of some individual risk in the event of system crisis. The first-order TM and the second-order TCM coincide with the popular risk measures called the marginal expected shortfall and the tail variance, respectively. We derive asymptotic expressions for the TM and TCM with any positive integer orders, when the individual risks are pairwise asymptotically independent and have distributions from certain classes that contain both light-tailed and heavy-tailed distributions. The formulas obtained possess concise forms unrelated to dependence structures, and hence enable us to estimate the TM and TCM efficiently. To demonstrate the wide application of our results, we revisit some issues related to premium principles and optimal capital allocation from the asymptotic point of view. We also give a numerical study on the relative errors of the asymptotic results obtained, under some specific scenarios when there are two individual risks in the system. The corresponding asymptotic properties of the degenerate univariate versions of the TM and TCM are discussed separately in an appendix at the end of the paper.

By considering the extreme behavior of bivariate random vectors with a polar representation R(u(T), v(T)), it is commonly assumed that the radial component R and the angular component T are stochastically independent. We investigate how to relax this rigid independence assumption such that conditional limit theorems can still be deduced. For this purpose, we introduce a novel measure for the dependence structure and present convenient criteria for validity of limit theorems possessing a geometrical meaning. Thus, our results verify a stability of the available limit results, which is essential in applications where the independence of the polar components is not necessarily present or exactly fulfilled.

In this paper we work in the framework of a k-dimensional vector of log-linear risks. Under weak conditions on the marginal tails and the dependence structure of a vector of positive risks, we derive the asymptotic tail behaviour of the aggregated risk and present an application concerning log-normal risks with stochastic volatility.