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Extremes of regularly varying Lévy-driven mixed moving average processes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Vicky Fasen
Vicky Fasen*
Affiliation:
Munich University of Technology
*
∗ Postal address: Graduate Program in Applied Algorithmic Mathematics, Centre for Mathematical Sciences, Munich University of Technology, D-85747 Garching, Germany. Email address: fasen@ma.tum.de
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Abstract

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In this paper, we study the extremal behavior of stationary mixed moving average processes of the form Y(t)=∫ℝ+×ℝf(r,t-s) dΛ(r,s), t∈ℝ, where f is a deterministic function and Λ is an infinitely divisible, independently scattered random measure whose underlying driving Lévy process is regularly varying. We give sufficient conditions for the stationarity of Y and compute the tail behavior of certain functionals of Y. The extremal behavior is modeled by marked point processes on a discrete-time skeleton chosen properly by the jump times of the underlying driving Lévy process and the extremes of the kernel function. The sequences of marked point processes converge weakly to a cluster Poisson random measure and reflect extremes of Y at a high level. We also show convergence of the partial maxima to the Fréchet distribution. Our models and results cover short- and long-range dependence regimes.

Keywords

Continuous-time moving average processextreme-value theoryindependently scattered random measurelong-range dependencemarked point processpoint processregular variationshot noise processsupOU process

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems 60G10: Stationary processes 60G55: Point processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 4 , December 2005 , pp. 993 - 1014
Copyright
Copyright © Applied Probability Trust 2005 

References

Albin, J. M. P. (2000). Extremes and upcrossing intensities for P-differentiable stationary processes. Stoch. Process. Appl. 87, 199–234.Google Scholar
Barndorff-Nielsen, O. E. (2001). Superposition of Ornstein–Uhlenbeck type processes. Theory Prob. Appl. 45, 175–194.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Modelling by Lévy processes for financial econometrics. In Lévy Processes, eds Barndorff-Nielsen, O. E., Mikosch, T. and Resnick, S. I., Birkhäuser, Boston, MA, pp. 283–318.CrossRefGoogle Scholar
Basrak, B., Davis, R. A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Prob. 12, 908–920.Google Scholar
Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Braverman, M. and Samorodnitsky, G. (1995). Functionals of infinitely divisible stochastic processes with exponential tails. Stoch. Process. Appl. 56, 207–231.Google Scholar
Breiman, L. (1965). On some limit theorems similar to the arc-sine law. Theory Prob. Appl. 10, 323–331.Google Scholar
Brockwell, P. J. and Marquardt, T. (2005). Lévy-driven and fractionally integrated ARMA processes with continuous time parameter. Statist. Sinica 15, 477–494.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. 1, Elementary Theory and Methods, 2nd edn. Springer, New York.Google Scholar
Davis, R. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Prob. 23, 879–917.Google Scholar
Davis, R. and Mikosch, T. (1998). The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26, 2049–2080.Google Scholar
Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179–195.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.Google Scholar
Fasen, V. (2004). Extremes of Lévy driven MA processes with applications in finance. , Munich University of Technology. Available at http://tumb1.biblio.tu-muenchen.de/publ/diss/ma/2004/fasen.html.Google Scholar
Fasen, V. (2005). Extremes of subexponential Lévy driven moving average processes. Submitted.Google Scholar
Gushchin, A. A. and Küchler, U. (2000). On stationary solutions of delay differential equations driven by a Lévy process. Stoch. Process. Appl. 88, 195–211.Google Scholar
Hsing, T. (1993). On some estimates based on sample behavior near high level excursions. Prob. Theory Relat. Fields 95, 331–356.Google Scholar
Hsing, T. and Teugels, J. L. (1989). Extremal properties of shot noise processes. Adv. Appl. Prob. 21, 513–525.Google Scholar
Hult, H. and Lindskøg, F. (2005). Extremal behavior of regularly varying stochastic processes. Stoch. Process. Appl. 115, 249–274.Google Scholar
Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.Google Scholar
Kwapień, S. and Woyczyński, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston, MA.Google Scholar
Leadbetter, M. R. and Rootzén, H. (1988). Extremal theory for stochastic processes. Ann. Prob. 16, 431–478.Google Scholar
Lebedev, A. V. (2000). Extremes of subexponential shot noise. Math. Notes 71, 206–210.Google Scholar
Lindskøg, F. (2004). Multivariate extremes and regular variation for stochastic processes. , ETH Zürich. Available at http://e-collection.ethbib.ethz.ch/cgi-bin/show.pl?type=diss&nr=15319.Google Scholar
McCormick, W. P. (1997). Extremes for shot noise processes with heavy tailed amplitudes. J. Appl. Prob. 34, 643–656.Google Scholar
Mikosch, T. and Samorodnitsky, G. (2000). The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Prob. 10, 1025–1064.CrossRefGoogle Scholar
Pedersen, J. (2003). The Lévy–Ito decomposition of an independently scattered random measure. Tech. Rep. 2003-2, Centre for Mathematical Physics and Stochastics, University of Aarhus. Available at http://www.maphysto.dk/oldpages/publications/publications2003_static.html.Google Scholar
Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 453–487.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Rootzén, H. (1978). Extremes of moving averages of stable processes. Ann. Prob. 6, 847–869.Google Scholar
Rootzén, H. (1986). Extreme value theory for moving average processes. Ann. Prob. 14, 612–652.Google Scholar
Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Prob. 21, 996–1014.Google Scholar
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Urbanic, K. and Woyczyński, W. A. (1968). Random measures and harmonizable sequences. Studia Math. 31, 61–88.Google Scholar