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.

The extremal behaviour of a Markov chain is typically characterised by its tail chain. For asymptotically dependent Markov chains, existing formulations fail to capture the full evolution of the extreme event when the chain moves out of the extreme tail region, and, for asymptotically independent chains, recent results fail to cover well-known asymptotically independent processes, such as Markov processes with a Gaussian copula between consecutive values. We use more sophisticated limiting mechanisms that cover a broader class of asymptotically independent processes than current methods, including an extension of the canonical Heffernan‒Tawn normalisation scheme, and reveal features which existing methods reduce to a degenerate form associated with nonextreme states.

Multivariate regular variation plays a role in assessing tail risk in diverse applications such as finance, telecommunications, insurance, and environmental science. The classical theory, being based on an asymptotic model, sometimes leads to inaccurate and useless estimates of probabilities of joint tail regions. This problem can be partly ameliorated by using hidden regular variation (see Resnick (2002) and Mitra and Resnick (2011)). We offer a more flexible definition of hidden regular variation that provides improved risk estimates for a larger class of tail risk regions.

Let {Xt, t ≥ 1} be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails, and let {Θt, t ≥ 1} be a sequence of positive random variables independent of the sequence {Xt, t ≥ 1}. We will discuss the tail probabilities and almost-sure convergence of X(∞) = ∑t=1∞ΘtXt+ (where X+ = max{0, X}) and max1≤k<∞∑t=1kΘtXt, and provide some sufficient conditions motivated by Denisov and Zwart (2007) as alternatives to the usual moment conditions. In particular, we illustrate how the conditions on the slowly varying function involved in the tail probability of X1 help to control the tail behavior of the randomly weighted sums. Note that, the above results allow us to choose X1, X2,… as independent and identically distributed positive random variables. If X1 has a regularly varying tail of index -α, where α > 0, and if {Θt, t ≥ 1} is a positive sequence of random variables independent of {Xt}, then it is known – which can also be obtained from the sufficient conditions in this article – that, under some appropriate moment conditions on {Θt, t ≥ 1}, X(∞) = ∑t=1∞ΘtXt converges with probability 1 and has a regularly varying tail of index -α. Motivated by the converse problems in Jacobsen, Mikosch, Rosiński and Samorodnitsky (2009) we ask the question: if X(∞) has a regularly varying tail then does X1 have a regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including the nonvanishing Mellin transform of ∑t=1∞Θt along some vertical line in the complex plane, so that the above is true. We also show that the condition on the Mellin transform cannot be dropped.

A two-dimensional random vector in the domain of attraction of an extreme value distribution G is said to be asymptotically independent (i.e. in the tail) if G is the product of its marginal distribution functions. Ledford and Tawn (1996) discussed a form of residual dependence in this case. In this paper we give a characterization of this phenomenon (see also Ramos and Ledford (2009)), and offer extensions to higher-dimensional spaces and stochastic processes. Systemic risk in the banking system is treated in a similar framework.

Asymptotic independence of the components of random vectors is a concept used in many applications. The standard criteria for checking asymptotic independence are given in terms of distribution functions (DFs). DFs are rarely available in an explicit form, especially in the multivariate case. Often we are given the form of the density or, via the shape of the data clouds, we can obtain a good geometric image of the asymptotic shape of the level sets of the density. In this paper we establish a simple sufficient condition for asymptotic independence for light-tailed densities in terms of this asymptotic shape. This condition extends Sibuya's classic result on asymptotic independence for Gaussian densities.

We study the tail behavior of the distribution of the sum of asymptotically independent risks whose marginal distributions belong to the maximal domain of attraction of the Gumbel distribution. We impose conditions on the distribution of the risks (X, Y) such that P(X + Y > x) ∼ (constant) P(X > x). With the further assumption of nonnegativity of the risks, the result is extended to more than two risks. We note a sufficient condition for a distribution to belong to both the maximal domain of attraction of the Gumbel distribution and the subexponential class. We provide examples of distributions which satisfy our assumptions. The examples include cases where the marginal distributions of X and Y are subexponential and also cases where they are not. In addition, the asymptotic behavior of linear combinations of such risks with positive coefficients is explored, leading to an approximate solution of an optimization problem which is applied to portfolio design.

Random vectors in the positive orthant whose distributions possess hidden regular variation are a subclass of those whose distributions are multivariate regularly varying with asymptotic independence. The concept is an elaboration of the coefficient of tail dependence of Ledford and Tawn. We show that the rank transform that brings unequal marginals to the standard case also preserves the hidden regular variation. We discuss applications of the results to two examples, one involving flood risk and the other Internet data.

Univariate heavy-tailed analysis rests on the analytic notion of regularly varying functions. For multivariate heavy-tailed analysis, reliance on functions is awkward because multivariate distribution functions are not natural objects for many purposes and are difficult to manipulate. An approach based on vague convergence of measures makes the differences between univariate and multivariate analysis evaporate. We survey the foundations of the subject and discuss statistical attempts to assess dependence of large values. An exploratory technique is applied to exchange rate return data and shows clear differences in the dependence structure of large values for the Japanese Yen versus German Mark compared with the French Franc versus the German Mark.

In this paper we study the asymptotic behaviour of the joint distribution of reinsurance aggregate claim amounts for large values of the retention level under various dependence assumptions. We prove that, under certain dependence assumptions, for large values of the retention level the ratio between the joint distribution of the aggregate losses and the product of the marginal distributions converges to a constant value that only depends on the frequency parameters.

The usual concept of asymptotic independence, as discussed in the context of extreme value theory, requires the distribution of the coordinatewise sample maxima under suitable centering and scaling to converge to a product measure. However, this definition is too broad to conclude anything interesting about the tail behavior of the product of two random variables that are asymptotically independent. Here we introduce a new concept of asymptotic independence which allows us to study the tail behavior of products. We carefully discuss equivalent formulations of asymptotic independence. We then use the concept in the study of a network traffic model. The usual infinite source Poisson network model assumes that sources begin data transmissions at Poisson time points and continue for random lengths of time. It is assumed that the data transmissions proceed at a constant, nonrandom rate over the entire length of the transmission. However, analysis of network data suggests that the transmission rate is also random with a regularly varying tail. So, we modify the usual model to allow transmission sources to transmit at a random rate over the length of the transmission. We assume that the rate and the time have finite mean, infinite variance and possess asymptotic independence, as defined in the paper. We finally prove a limit theorem for the input process showing that the centered cumulative process under a suitable scaling converges to a totally skewed stable Lévy motion in the sense of finite-dimensional distributions.

The influence of bivariate extremal dependence on the limiting behaviour of the concomitant of the largest order statistic is examined. Our approach is to fix the marginal distributions and derive a general tail characterisation of the joint survivor function. From this, we identify the normalisation required to obtain the limiting distribution of the concomitant of the largest order statistic, obtain its tail form, and investigate the limiting probability that the vector of componentwise maxima occurs as an observation of the bivariate process. The results are illustrated for a range of extremal dependence forms.