Albrecher, Hansjörg; Asmussen, Søren; Kortschak, Dominik Tail asymptotics for the sum of two heavy-tailed dependent risks. (English) Zbl 1142.60009 Extremes 9, No. 2, 107-130 (2006). For a heavy-tailed bivariate r.v. \((X_1,X_2)\) the tail behaviour of \(X_1+X_2\) is investigated in a general copula framework. Representations and inequalities for \(P(X_1+X_2>x)\) are given. E.g. it is shown that if \(P(X_1>x)\) is regularly varying with an index \(-\alpha\), then \( \lim\sup_{x\to\infty} P(X_1+X_2>x)/P(X_1>x) \) is less then \((\hat\lambda^{1/(\alpha+1)}+(1+c-2\hat\lambda))^{\alpha+1}\) if \(\hat\lambda\leq (1+c)/3\), where \(\hat\lambda=\lim_{x\to\infty}P(X_2>x| X_1>x)\), \(c=\lim_{x\to\infty}P(X_2>x)/P(X_1>x)<1\). Lognormal marginal distribution, Archimedian, Farlie-Gumbel-Morgenstern and linear Spearman copulas are considered as examples. Reviewer: R. E. Maiboroda (Kyïv) Cited in 36 Documents MSC: 60E05 Probability distributions: general theory 62E20 Asymptotic distribution theory in statistics Keywords:copula; mean excess function; subexponential distribution; tail dependence PDFBibTeX XMLCite \textit{H. Albrecher} et al., Extremes 9, No. 2, 107--130 (2006; Zbl 1142.60009) Full Text: DOI References: [1] Abdous, B., Ghoudi, K., Khoudraji, A.: Non-parametric estimation of the limit dependence function of multivariate extremes. Extremes 2(3), 245–268 (1999) · Zbl 0957.62044 [2] Alink, S., Löwe, M., Wüthrich, M.V.: Diversification of aggregate dependent risks. Insur. Math. Econ. 35(1), 77–95 (2004) · Zbl 1052.62105 [3] Asmussen, S.: Ruin Probabilities. World Scientific, Singapore (2000) · Zbl 0960.60003 [4] Asmussen, S., Rojas-Nandaypa, L.: Sums of dependent lognormal random variables: asymptotics and simulation. Preprint (2005) [5] Basrak, B., Davis, R.A., Mikosch, T.: A characterization of multivariate regular variation. Ann. Appl. Probab. 12(3), 908–920 (2002) · Zbl 1070.60011 [6] Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1989) · Zbl 0667.26003 [7] Coles, S., Heffernan, J., Tawn, J.: Dependence measures for extreme value analyses. Extremes 2(4), 339–365 (1999) · Zbl 0972.62030 [8] Cossette, H., Denuit, M., Marceau, É.: Distributional bounds for functions of dependent risks. Schweiz. Aktuarver. Mitt. (1), 45–65 (2002) · Zbl 1187.91093 [9] Davis, R.A., Resnick, S.I.: Limit theory for bilinear processes with heavy-tailed noise. Ann. Appl. Probab. 6(4), 1191–1210 (1996) · Zbl 0879.60053 [10] Denuit, M., Genest, C., Marceau, É.: Stochastic bounds on sums of dependent risks. Insur., Math. Econ. 25(1), 85–104 (1999) · Zbl 1028.91553 [11] Embrechts, P., Puccetti, G.: Bounds for functions of dependent risks. Preprint (2005) · Zbl 1101.60010 [12] Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events. Springer, Berlin Heidelberg New York (1997) · Zbl 0873.62116 [13] Frahm, G., Junker, M., Schmidt, R.: Estimating the tail-dependence coefficient: properties and pitfalls. Insur. Math. Econ. 37(1), 80–100 (2005) · Zbl 1101.62012 [14] Geluk, J., Ng, K.: Tail behavior of negatively associated heavy tailed sums. J. Appl. Probab. 43(2), 587–593 (2006) · Zbl 1104.60313 [15] Goovaerts, M., Kaas, R., Tang, Q., Vernic, R.: The tail probability of discounted sums of pareto-like losses in insurance. Proceedings of the 8th Int. Congress on Insurance: Mathematics & Economics, Rome (2004) · Zbl 1144.91026 [16] Hult, H., Lindskog, F.: Multivariate extremes, aggregation and dependence in elliptical distributions. Adv. Appl. Probab. 34(3), 587–608 (2002) · Zbl 1023.60021 [17] Joe. H.: Multivariate Models and Dependence Concepts. Chapman & Hall, London (1997) · Zbl 0990.62517 [18] Juri, A., Wüthrich, M.V.: Copula convergence theorems for tail events. Insur. Math. Econ. 30(3), 405–420 (2002) · Zbl 1039.62043 [19] Juri, A., Wüthrich, M.V.: Tail dependence from a distributional point of view. Extremes 6(3), 213–246 (2003) · Zbl 1049.62055 [20] Ledford, A.W., Tawn, J.A.: Modelling dependence within joint tail regions. J. R. Stat. Soc. Ser. B 59(2), 475–499 (1997) · Zbl 0886.62063 [21] Makarov, G.D.: Estimates for the distribution function of the sum of two random variables with given marginal distributions. Theory Probab. Appl. 26(4), 803–806 (1981) · Zbl 0488.60022 [22] Malevergne, Y., Sornette, D.: Investigating extreme dependences: concepts and tools. Review of Financial Studies (2006). (to appear). · Zbl 1093.62098 [23] Mesfioui, M., Quessy, J.F.: Bounds on the value-at-risk for the sum of possibly dependent risks. Insur. Math. Econ. 37, 135–151 (2005) · Zbl 1115.91032 [24] Nelsen, R.: An Introduction to Copulas. Springer, Berlin Heidelberg New York (1999) · Zbl 0909.62052 [25] Pratt, J.W.: On interchanging limits and integrals. Ann. Math. Stat. 31, 74–77 (1960) · Zbl 0090.26802 [26] Resnick, S.: Extreme Values, Regular Variation, and Point Processes. Springer, Berlin Heidelberg New York (1987) · Zbl 0633.60001 [27] Resnick, S.: Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5(4), 303–336 (2002) · Zbl 1035.60053 [28] Resnick, S.: The extremal dependence measure and asymptotic independence. Stoch. Models 20(2), 205–227 (2004) · Zbl 1054.62063 [29] Schmidt, R., Stadtmüller, U.: Non-parametric estimation of tail dependence. Scand. J. Statist. 33, 307–335 (2006) · Zbl 1124.62016 [30] Tang, Q., Wang, D.: Tail Probabilities of randomly weighted sums of random variables with dominated variation. Stoch. Models 22(2), 253–272 (2006) · Zbl 1095.60008 [31] Wang, D., Tang, Q.: Maxima of sums and random sums for negatively associated random variables with heavy tails. Stat. Probab. Lett. 68, 287–295 (2004) · Zbl 1116.62351 [32] Wüthrich, M.V.: Asymptotic value-at-risk estimates for sums of dependent random variables. Astin Bull. 33(1), 75–92 (2003) · Zbl 1098.62570 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.