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On regular variation for infinitely divisible random vectors and additive processes

Part of: Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Henrik Hult  and
Filip Lindskog
Henrik Hult*
Affiliation:
Cornell University
Filip Lindskog*
Affiliation:
KTH Stockholm
*
∗ Postal address: School of Operations Research and Industrial Engineering, Cornell University, 414A Rhodes Hall, Ithaca, NY 14853, USA. Email address: hh228@cornell.edu
∗∗ Postal address: Department of Mathematics, KTH, 100 44 Stockholm, Sweden.
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Abstract

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We study the tail behavior of regularly varying infinitely divisible random vectors and additive processes, i.e. stochastic processes with independent but not necessarily stationary increments. We show that the distribution of an infinitely divisible random vector is tail equivalent to its Lévy measure and we study the asymptotic decay of the probability for an additive process to hit sets far away from the origin. The results are extensions of known univariate results to the multivariate setting; we exemplify some of the difficulties that arise in the multivariate case.

Keywords

Multivariate regular variationinfinitely divisible distributionadditive processLevy process

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60G70: Extreme value theory; extremal processes 60E07: Infinitely divisible distributions; stable distributions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 134 - 148
Copyright
Copyright © Applied Probability Trust 2006 

References

Basrak, B. (2000). The sample autocorrelation function of non-linear time series. , Department of Mathematics, University of Groningen.Google Scholar
Basrak, B., Davis, R. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Prob. 12, 908–920.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
Braverman, M., Mikosch, T. and Samorodnitsky, G. (2002). Tail probabilities of subadditive functionals acting on Lévy processes. Ann. Appl. Prob. 12, 69–100.Google Scholar
Cline, D. and Resnick, S. I. (1992). Multivariate subexponential distributions. Stoch. Process. Appl. 42, 49–72.Google Scholar
Embrechts, P., Goldie, C. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.Google Scholar
De Haan, L. and Lin, T. (2001). On convergence toward an extreme value distribution in C0,1. Ann. Prob. 29, 467–483.Google Scholar
Hult, H. and Lindskog, F. (2005). Extremal behavior for regularly varying stochastic processes. Stoch. Process. Appl. 115, 249–274.Google Scholar
Hult, H. and Lindskog, F. (2006). Extremal behavior of stochastic integrals driven by regularly varying Lévy processes. To appear in Ann. Prob. Google Scholar
Hult, H. and Lindskog, F. (2006). On Kesten's counterexample to the Cramér–Wold device for regular variation. Bernoulli 12, 133–142.Google Scholar
Hult, H., Lindskog, F., Mikosch, T. and Samorodnitsky, G. (2005). Functional large deviations for multivariate regularly varying random walks. Ann. Appl. Prob. 15, 2651–2680.Google Scholar
Kallenberg, O. (1983). Random Measures, 3rd edn. Akademie-Verlag, Berlin.Google Scholar
Lindskog, F. (2004). Multivariate extremes and regular variation for stochastic processes. , Department of Mathematics, Swiss Federal Institute of Technology, Zürich. Available at http://e-collection.ethbib.ethz.ch/show?type=diss&nr=15319.Google Scholar
Pratt, J. (1960). On interchanging limits and integrals. Ann. Math. Statist. 31, 74–77.Google Scholar
Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
Resnick, S. (2004). On the foundations of multivariate heavy-tail analysis. In Stochastic Methods and Their Applications (J. Appl. Prob. Spec. Vol. 41A), Applied Probability Trust, Sheffield, pp. 191–212.Google Scholar
Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. Appl. Prob. 18, 66–138.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Prob. 21, 993–1014.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions (Camb. Stud. Adv. Math. 68). Cambridge University Press.Google Scholar
Willekens, E. (1987). On the supremum of an infinitely divisible process. Stoch. Process. Appl. 26, 173–175.Google Scholar