Extremes of aggregated Dirichlet risks

Abstract

a b s t r a c tThe class of Dirichlet random vectors is central in numerous probabilistic and statistical applications. The main result of this paper derives the exact tail asymptotics of the aggregated risk of powers of Dirichlet random vectors when the radial component has df in the Gumbel or the Weibull max-domain of attraction. We present further results for the joint asymptotic independence and the max-sum equivalence.

Extremal attractors of Liouville copulas

“…Theorem 1 specifies when X ∈ M(H 0 ) and identifies H 0 . While part (a) follows from regular variation of X, parts (b) and (c) are special cases of the results discussed in Section 2.2 in [19]. Details of the proof may be found in B.…”

“…As a byproduct, we also obtain the lower and upper tail dependence coefficients of Liouville copulas that quantify the strength of dependence at extreme levels [25]. These results are complementary to [21], where the upper tail order functions of a Liouville copula and its density are derived when α 1 = · · · = α d , and to [19], where the extremal attractor of RD α is derived when R is light-tailed. The extremal attractors of Liouville copulas are interesting in their own right.…”

“…The part (b) follows directly from Proposition 2.2 in [19] upon setting p = 1 and taking, for i = 1, . .…”

“…(B.1) and hence also Theorem 1 (c). Note that alternatively, part (c) could be proved using Theorem 2.1 in [19] similarly to the proof of Proposition 2.2 therein.…”

Asymptotic expansions for bivariate normal extremes

“…However, there have been studies on convergence aspects of the extremes of other multivariate distributions and multivariate processes, see Hashorva andJi (2014a, 2014b), Hashorva and Kortschak (2014), Hashorva (2015), Hashorva and Ji (2015), , Hashorva and Li (2015), Hashorva and Ji (2016), Hashorva and Ling (2016) and .…”