Advances in Applied Probability

Extremes of regularly varying Lévy-driven mixed moving average processes

Vicky Fasen

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In this paper, we study the extremal behavior of stationary mixed moving average processes of the form Y(t)=∫+×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.

Article information

Adv. in Appl. Probab. Volume 37, Number 4 (2005), 993-1014.

First available in Project Euclid: 14 December 2005

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Mathematical Reviews number (MathSciNet)

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems 60G10: Stationary processes 60G55: Point processes

Continuous-time moving average process extreme-value theory independently scattered random measure long-range dependence marked point process point process regular variation shot noise process supOU process


Fasen, Vicky. Extremes of regularly varying Lévy-driven mixed moving average processes. Adv. in Appl. Probab. 37 (2005), no. 4, 993--1014. doi:10.1239/aap/1134587750.

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  • Albin, J. M. P. (2000). Extremes and upcrossing intensities for $P$-differentiable stationary processes. Stoch. Process. Appl. 87, 199--234.
  • Barndorff-Nielsen, O. E. (2001). Superposition of Ornstein--Uhlenbeck type processes. Theory Prob. Appl. 45, 175--194.
  • Barndorff-Nielsen, O. E. and Shephard, N. (2001). Modelling by Lévy processes for financial econometrics. In Lévy Processes, eds O. E. Barndorff-Nielsen, T. Mikosch and S. I. Resnick, Birkhäuser, Boston, MA, pp. 283--318.
  • Basrak, B., Davis, R. A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Prob. 12, 908--920.
  • Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95--115.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.
  • Braverman, M. and Samorodnitsky, G. (1995). Functionals of infinitely divisible stochastic processes with exponential tails. Stoch. Process. Appl. 56, 207--231.
  • Breiman, L. (1965). On some limit theorems similar to the arc-sine law. Theory Prob. Appl. 10, 323--331.
  • Brockwell, P. J. and Marquardt, T. (2005). Lévy-driven and fractionally integrated ARMA processes with continuous time parameter. Statist. Sinica 15, 477--494.
  • 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.
  • 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.
  • Davis, R. and Mikosch, T. (1998). The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26, 2049--2080.
  • Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179--195.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
  • Fasen, V. (2004). Extremes of Lévy driven MA processes with applications in finance. Doctoral Thesis, Munich University of Technology. Available at
  • Fasen, V. (2005). Extremes of subexponential Lévy driven moving average processes. Submitted. %Preprint, available
  • 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.
  • Hsing, T. (1993). On some estimates based on sample behavior near high level excursions. Prob. Theory Relat. Fields 95, 331--356.
  • Hsing, T. and Teugels, J. L. (1989). Extremal properties of shot noise processes. Adv. Appl. Prob. 21, 513--525.
  • Hult, H. and Lindskøg, F. (2005). Extremal behavior of regularly varying stochastic processes. Stoch. Process. Appl. 115, 249--274.
  • Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
  • Kwapień, S. and Woyczyński, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston, MA.
  • Leadbetter, M. R. and Rootzén, H. (1988). Extremal theory for stochastic processes. Ann. Prob. 16, 431--478.
  • Lebedev, A. V. (2000). Extremes of subexponential shot noise. Math. Notes 71, 206--210.
  • Lindskøg, F. (2004). Multivariate extremes and regular variation for stochastic processes. Doctoral Thesis, ETH Zürich. Available at
  • McCormick, W. P. (1997). Extremes for shot noise processes with heavy tailed amplitudes. J. Appl. Prob. 34, 643--656.
  • 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.
  • 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.
  • Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 453--487.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • Rootzén, H. (1978). Extremes of moving averages of stable processes. Ann. Prob. 6, 847--869.
  • Rootzén, H. (1986). Extreme value theory for moving average processes. Ann. Prob. 14, 612--652.
  • Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Prob. 21, 996--1014.
  • Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
  • Urbanic, K. and Woyczyński, W. A. (1968). Random measures and harmonizable sequences. Studia Math. 31, 61--88.