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Pareto Lévy measures and multivariate regular variation. (English) Zbl 1248.60052
The authors consider regular variation of an \(\mathbb R^d\)-valued Lévy process \(X\) with Lévy measure \(\Pi\), emphasizing the dependence between jumps of its components. By transforming the marginal Lévy measures to those of a standard \(1\)-stable Lévy process, they decouple the marginal Lévy measure from the dependence structure. The dependence between the jumps is modeled by the so-called Pareto Lévy measure. The authors characterize the multivariate regular variation of \(X\) by its one-dimensional marginal Lévy measures and the Pareto Lévy measure.

MSC:
60G51 Processes with independent increments; Lévy processes
60E07 Infinitely divisible distributions; stable distributions
60G52 Stable stochastic processes
60G70 Extreme value theory; extremal stochastic processes
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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[1] Barndorff-Nielsen, O. E. and Lindner, A. M. (2007). Lévy copulas: dynamics and transforms of Upsilon type. Scand. J. Statist. 34, 298-316. · Zbl 1141.60022
[2] Basrak, B. (2000). The sample autocorrelation function of non-linear time series. Doctoral Thesis, Rijksuniversiteit Groningen.
[3] Basrak, B., Davis, R. A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Prob. 12, 908-920. · Zbl 1070.60011
[4] Böcker, K. and Klüppelberg, C. (2010). Multivariate models for operational risk. Quant. Finance 10 , 855-869. · Zbl 1204.91059
[5] Bregman, Y. and Klüppelberg, C. (2005). Ruin estimation in multivariate models with Clayton dependence structure. Scand. Actuarial J. 2005, 462-480. · Zbl 1145.91031
[6] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes . Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1052.91043
[7] De Haan, L. and Lin, T. (2001). On convergence toward an extreme value distribution in \(C[0,1]\). Ann. Prob. 29, 467-483. · Zbl 1010.62016
[8] Eder, I. (2009). First passage events and multivariate regular variation for dependent Lévy processes with applications in insurance. Doctoral Thesis, Technische Universität München.
[9] Eder, I. and Klüppelberg, C. (2009). The first passage event for sums of dependent Lévy processes with applications to insurance risk. Ann. Appl. Prob. 19, 2047-2079. · Zbl 1209.60029
[10] Esmaeili, H. and Klüppelberg, C. (2010). Parameter estimation of a bivariate compound Poisson process. Insurance Math. Econom. 47, 224-233. · Zbl 1231.62150
[11] Esmaeili, H. and Klüppelberg, C. (2011). Parametric estimation of a bivariate stable Lévy process J. Multivariate Anal. 102, 918-930. · Zbl 1210.62111
[12] Esmaeili, H. and Klüppelberg, C. (2011). Two-step estimation of a multivariate Lévy process. Submitted. · Zbl 1210.62111
[13] Giné, E., Hahn, M. G. and Vatan, P. (1990). Max-infinitely divisible and max-stable sample continuous processes. Prob. Theory Relat. Fields 87, 139-165. · Zbl 0688.60031
[14] Hult, H. and Lindskog, F. (2005). Extremal behavior of regularly varying stochastic processes. Stoch. Process. Appl. 115, 249-274. · Zbl 1070.60046
[15] Hult, H. and Lindskog, F. (2006). On regular variation for infinitely divisible random vectors and additive processes. Adv. Appl. Prob. 38, 134-148. · Zbl 1106.60046
[16] Hult, H. and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. 80, 121-140. · Zbl 1164.28005
[17] Hult, H. and Lindskog, F. (2007). Extremal behavior of stochastic integrals driven by regularly varying Joe, H. (1997). Multivariate Models and Dependence Concepts . Chapman & Hall/CRC, London. · Zbl 0990.62517
[18] Kallenberg, O. (1983). Random Measures , 3rd edn. Akademie, Berlin. · Zbl 0544.60053
[19] Kallsen, J. and Tankov, P. (2006). Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J. Multivariate Anal. 97, 1551-1572. · Zbl 1099.62048
[20] Klüppelberg, C. and Resnick, S. I. (2008). The Pareto copula, aggregation of risks, and the emperor’s socks. J. Appl. Prob. 45, 67-84. · Zbl 1144.62037
[21] Nelsen, R. (2006). An Introduction to Copulas , 2nd edn. Springer, New York. · Zbl 1152.62030
[22] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes . Springer, New York. · Zbl 0633.60001
[23] Resnick, S. I. (2007). Heavy-Tail Phenomena . Springer, New York. · Zbl 1152.62029
[24] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes . Chapman & Hall, New York. · Zbl 0925.60027
[25] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions . Cambridge University Press. · Zbl 0973.60001
[26] Ueltzhöfer, F. A. J. and Klüppelberg, C. (2011). An oracle inequality for penalised projection estimation of Lévy densities from high-frequency observations. J. Nonparametric Statist. 23, 967-989. · Zbl 1229.62108
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