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Tail asymptotics for the sum of two heavy-tailed dependent risks. (English) Zbl 1142.60009

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.

MSC:

60E05 Probability distributions: general theory
62E20 Asymptotic distribution theory in statistics
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[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
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