Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T04:59:38.327Z Has data issue: false hasContentIssue false
  • English
  • Français

Extreme values of the cyclostationary Gaussian random process

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

D. G. Konstant  and
V.I. Piterbarg
D. G. Konstant
Affiliation:
National Technical University, Athens
V.I. Piterbarg*
Affiliation:
Moscow State University
*
∗∗ Postal address: Department of Probability Theory, Faculty of Mathematics (MexMat), Moscow State University (MGU), Moscow 117234, Russia.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper the class of cyclostationary Gaussian random processes is studied. Basic asymptotics are given for the class of Gaussian processes that are centered and differentiable in mean square. Then, under certain conditions on the non-degeneration of the centered cyclostationary Gaussian process with integrable covariance functions, the Gnedenko-type limit formula is established for and all x > 0.

Keywords

PERIODICALLY CORRELATED PROCESSLARGE DEVIATIONS

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G17: Sample path properties
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 1 , March 1993 , pp. 82 - 97
Copyright
Copyright © Applied Probability Trust 1993 

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.)

Footnotes

∗

Present address: Department of Mathematics, Technical University of Crete, Chania 73100, Greece.

References

Adler, R. J. (1981) The Geometry of Random Fields. Wiley, New York.Google Scholar
Anderson, T. W. (1958) An Introduction to Multivariate Statistical Analysis. Wiley, New York.Google Scholar
Belyaev, Yu. K. and Piterbarg, V. I. (1972) Asymptotic of the mean number of A-points of excursions above high level of Gaussian field. In Excursions of Random Fields, pp. 62–89, Moscow University Press.Google Scholar
Bickel, P. J. and Rosenblatt, M. (1973) On some global measures of the deviations of density function estimates. Ann. Statist. 1, 1071–1095.CrossRefGoogle Scholar
Borell, C. (1975) The Brunn Minkowski inequality in Gauss space. Invent. Math. 30, 207–216.CrossRefGoogle Scholar
Cramer, H. and Leadbetter, M.R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Dobric, V., Marcus, M. B. and Weber, M. J. G. (1988) The distribution of large values of the supremum of a Gaussian process. Astérisque, 157–158, 95-127.Google Scholar
Fatalov, V.R. (1983) Exact asymptotics of distribution functions of the maximum Gaussian nonhomogeneous random field. Dokl. Acad. Sci. Armenian SSR, 1, 25–29.Google Scholar
Gnedenko, B. V. (1943) Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44, 423–453.CrossRefGoogle Scholar
Landau, H. J. and Shepp, L. A. (1971) On the supremum of a Gaussian process. Sankhya A32, 369–378.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1986) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.Google Scholar
Piterbarg, V. I. (1988) Theory of Asymptotic Methods in Gaussian Random Processes and Fields. Moscow University Press.Google Scholar
Piterbarg, V. I. and Prisyazhnyuk, V. P. (1981) Exact asymptotic of the probability of large deviation of Gaussian stationary process. Prob. Theory Appl. 26, 480–495.Google Scholar
Talagrand, M. (1988) Small tails for the supremum of a Gaussian process. Ann. Inst. H. Poincaré 24, 307–315.Google Scholar
Yaglom, A. M. (1987) Correlation Theory of Stationary and Related Random Functions, Volume 1: Basic Results; Volume 2: Notes and References. Springer-Verlag, New York.Google Scholar