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Living on the multidimensional edge: Seeking hidden risks using regular variation. (English) Zbl 1276.60041
The authors consider a nonnegative random vector which they interpret as a risk vector. The distribution of the vector is supposed to have multivariate regular variation that is an asymptotic relation which is called $$M^*$$-convergence. The notion of multivariate regular variation supplies tail equivalence of the marginal components, so it has to be modified for applications. But its advantage is that it allows the limit measure to be used for the approximation of the tail probabilities. Such approximations of tail probabilities are sensitive to degeneracies in the limit measure. Other advantages and disadvantages of this approach comparing to hidden regular variation (HRV) are analyzed. The $$M^*$$-convergence is reviewed and the reasons to abandon the standard practice of defining regular variation through vague convergence on a compactification of $$\mathbb{R}^d$$ are discussed. A nonstandard regular variation is considered from the point of view of regular variation on the sequence of cones. Nonstandard regular variation is essential in practice since it is not natural to assume in applications that all marginals in a multivariate model are tail equivalent. In the conclusion, it is discussed how to fit the model of HRV on cones to data as well as estimation techniques of tail probabilities using the model introduced in the paper.

##### MSC:
 60F99 Limit theorems in probability theory 62G32 Statistics of extreme values; tail inference 60G70 Extreme value theory; extremal stochastic processes
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