Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-29T00:23:15.712Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes

Published online by Cambridge University Press:  14 July 2016

O. V. Seleznjev
O. V. Seleznjev*
Affiliation:
Moscow State University
*
∗Postal address: Faculty of Mathematics and Mechanics, Moscow State University, 117234, Moscow, USSR.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.

Keywords

MAXIMUM OF GAUSSIAN PROCESSPOINT PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 1 , March 1991 , pp. 17 - 32
Copyright
Copyright © Applied Probability Trust 1991 

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

Part of this work was carried out during a visit to the Department of Mathematical Statistics, the University of Lund, Sweden.

References

Adler, R. J. (1989) An introduction to continuity, extrema, and related topics for general Gaussian processes. Preprint.Google Scholar
Belyaev, Ju. K. and Simonjan, A. H. (1977) Asymptotic properties of deviations of the path of Gaussian process from approximation broken line for decreasing width of quantization. In Random Processes and Fields, ed. Belyaev, Yu. K., Izdat. Mosk. Univ., Moscow, 9–21.Google Scholar
Dudley, R. M. (1973) Sample functions of the Gaussian processes. Ann. Prob. 1, 66–103.Google Scholar
Eplett, W. J. R. (1986) Approximation theory for simulation of continuous Gaussian processes. Prob. Theory Rel. Fields 73, 159–181.Google Scholar
Feinerman, R. P. and Newman, D. J. (1974) Polynomial Approximation. Williams and Wilkins, Baltimore.Google Scholar
Hasofer, A. M. (1987) Distribution of the maximum of a Gaussian process by a Monte Carlo method. J. Sound Vibr. 112, 283–293.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Pickands, J. III (1969) Upcrossings probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 51–73.Google Scholar
Piterbarg, V. I. (1972a) About the paper of Pickands, J. III “Upcrossings probabilities for stationary Gaussian processes”. Vestn. Mosk. Univ. Ser. Math. Mec. 5, 25–30.Google Scholar
Piterbarg, V. I. (1972b) Asymptotic Poisson law for the number of high overshoots and the distribution of Gaussian homogeneous field. In Overshoots of Random Fields, ed. Belyaev, Yu. K., Izdat. Mosk. Univ., Moscow.Google Scholar
Piterbarg, V. I. (1988) Asymptotic Methods in the Theory of Gaussian Random Processes and Fields. Izdat. Mosk. Univ., Moscow.Google Scholar
Qualls, C. and Watanabe, H. (1972) Asymptotic properties of Gaussian process. Ann. Math. Statist. 43, 580–596.Google Scholar
Seleznjev, O. V. (1980a) Approximation of periodic Gaussian processes by trigonometric polynomials. Soviet Math. Dokl. 211, 29–33.Google Scholar
Seleznjev, O. V. (1980b) Approximation of periodic Gaussian process by random trigonometric polynomials. In Random Processes and Fields, ed. Belyaev, Yu. K., Izdat. Mosk. Univ., Moscow, 84–95.Google Scholar
Seleznjev, O. V. (1989) The best approximation of random processes and approximation of periodic random processes. Univ. Lund Stat. Res. Rep. 6, 1–32.Google Scholar
Weber, M. (1989) The supremum of Gaussian processes with constant variance. Prob. Theory Rel. Fields 81, 585–591.Google Scholar