Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-21T22:16:18.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Maxima of continuous-time stationary stable processes

Part of: Ergodic theory Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Gennady Samorodnitsky
Gennady Samorodnitsky*
Affiliation:
Cornell University
*
∗ Postal address: School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the suprema over compact time intervals of stationary locally bounded α-stable processes. The behaviour of these suprema as the length of the time interval increases turns out to depend significantly on the ergodic-theoretical properties of a flow generating the stationary process.

Keywords

Stable processstationary processergodic theorymaximaextreme value theorynon-singular flowdissipative flowconservative flowweak convergence

MSC classification

Primary: 60G10: Stationary processes 37A40: Nonsingular (and infinite-measure preserving) transformations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004 , pp. 805 - 823
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aaronson, J. (1997). An Introduction to Infinite Ergodic Theory (Mathematical Surveys Monogr. 50). American Mathematical Society, Providence, RI.Google Scholar
Adler, R. J. (1990). An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes (IMS Lectures Notes 12). Institute of Mathematical Statistics, Hayward, CA.Google Scholar
Albin, J. M. P. (1990). On extremal theory for stationary processes. Ann. Prob. 18, 92–128.CrossRefGoogle Scholar
Albin, J. M. P. (1992). On the general law of iterated logarithm with application to selfsimilar processes and to Gaussian processes in ℝn and Hilbert space. Stoch. Process. Appl. 41, 1–31.CrossRefGoogle Scholar
Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth and Brooks/Cole, Pacific Grove, CA.Google Scholar
Borell, C. (1975). The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30, 205–216.Google Scholar
Dudley, R. M. (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1, 290–330.CrossRefGoogle Scholar
Embrechts, P. and Maejima, M. (2002). Selfsimilar Processes. Princeton University Press.Google Scholar
Krengel, U. (1985). Ergodic Theorems. De Gruyter, Berlin.CrossRefGoogle Scholar
Kwapień, S. and Woyczyński, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.Google Scholar
Rosiński, J., (1995). On the structure of stationary stable processes. Ann. Prob. 23, 1163–1187.Google Scholar
Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Prob. 21, 996–1014.CrossRefGoogle Scholar
Samorodnitsky, G. (2004). Extreme value theory, ergodic theory, and the boundary between short memory and long memory for stationary stable processes. Ann. Prob. 32, 1438–1468.CrossRefGoogle Scholar
Samorodnitsky, G. and Taqqu, M. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.Google Scholar