Skip to main content
SpringerLink
Log in

Asymptotic Behavior of Point Processes of Exceeding the High Levels of Gaussian Stationary Sequence

  • Published:
Moscow University Mathematics Bulletin Aims and scope

Abstract

The paper studies the asymptotic behavior of point processes of exits of a Gaussian stationary sequence beyond a level tending to infinity more slowly than in the Poisson limit theorem for the number of exits. Convergence in variation of such point processes to a marked Poisson process is proved. The results of Yu.V. Prokhorov on the best approximation of the Bernoulli distribution by a mixture of Gaussian and Poisson distributions are applied. A.N. Kolmogorov proposed this problem in the early 1950s.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. V.I. Piterbarg, Twenty Lectures on Gaussian Processes (MTsNMO, Moscow, 2015).

  2. Y. Mittal and D. Ylvisaker, ‘‘Limit distributions for the maxima of stationary Gaussian processes,’’ Stochastic Processes Their Appl. 3, 1–18 (1975). doi 10.1016/0304-4149(75)90002-2

  3. V. I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields, Translations of Mathematical Monographs, Vol. 148 (American Mathematical Society, Providence, R.I., 2012). https://doi.org/10.1090/mmono/148

  4. Yu. V. Prokhorov, ‘‘Asymptotic behavior of the binomial distribution,’’ Usp. Mat. Nauk 8, 135–142 (1953).

  5. P. A. Kozulyaev, ‘‘Asymptotic analysis of one main formula of the probability theory,’’ Uch. Zap. Mosk. Univ. 15, 179–182 (1939).

  6. V. I. Bogachev, Foundations of the Theory of Measure, Vol. 1 (Regulyarnaya i Khaoticheskaya Dinamika, Moscow, 2003).

  7. S. I. Resnick, Extreme Values, Regular Variation and Point Processes, Springer Series in Operations Research and Financial Engineering (Springer, New York, 1987). https://doi.org/10.1007/978-0-387-75953-1

  8. S. M. Berman, Sojourns and Extremes of Stochastic Processes (CRC Press, London, 1992).

  9. G. Ben Arous, L. V. Bogachev, and S. A. Molchanov, ‘‘Limit theorems for sums of random exponentials,’’ Probab. Theory Relat. Fields 132, 579–612 (2005). https://doi.org/10.1007/s00440-004-0406-3

  10. S. Molchanov and V. Panov, ‘‘Limit theorems for the alloy-type random energy model,’’ Stochastics 91, 754–772 (2019). https://doi.org/10.1080/17442508.2018.1545841

Download references

ACKNOWLEDGMENTS

We thank A.V. Bulinskii, M.Ya. Kel’bert, and S.A. Molchanov for fruitful discussions.

Funding

The work is partially supported by the Institute of System Analysis of the Russian Academy of Sciences and International Laboratory for Stochastic Analysis and Its Applications of the HSE University.

Author information

Authors and Affiliations

  1. Chair of Probability Theory, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    V. I. Piterbarg

Authors
  1. V. I. Piterbarg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. I. Piterbarg.

Ethics declarations

The author of this work declares that he has no conflicts of interest.

Additional information

Translated by E. Oborin

Publisher’s Note.

Allerton Press remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Piterbarg, V.I. Asymptotic Behavior of Point Processes of Exceeding the High Levels of Gaussian Stationary Sequence. Moscow Univ. Math. Bull. 78, 291–297 (2023). https://doi.org/10.3103/S0027132223060050

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0027132223060050

Keywords:

Search

Navigation