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Methodology and Computing in Applied Probability

, Volume 21, Issue 4, pp 1183–1213 | Cite as

Approximation of Sojourn Times of Gaussian Processes

  • Krzysztof Dȩbicki
  • Zbigniew Michna
  • Xiaofan PengEmail author
Article

Abstract

We investigate the tail asymptotic behavior of the sojourn time for a large class of centered Gaussian processes X, in both continuous- and discrete-time framework. All results obtained here are new for the discrete-time case. In the continuous-time case, we complement the investigations of Berman (Commun Pure Appl Math 38(5):519–528, 1985a and Probab Theory Relat Fields 20(1):113–124, 1987) for non-stationary X. A by-product of our investigation is a new representation of Pickands constant which is important for Monte-Carlo simulations and yields a sharp lower bound for Pickands constant.

Keywords

Sojourn time Occupation time Exact asymptotics Gaussian process Locally stationary processes 

Mathematics Subject Classification (2010)

Primary 60G15; Secondary 60G70 

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Notes

Acknowledgments

The authors would like to thank Enkelejd Hashorva for his numerous remarks on all the steps of preparation of the manuscript and for ideas that led to the proof of Theorem 1.1. K.D. was partially supported by NCN Grant No 2015/17/B/ST1/01102 (2016-2019). X.P. thanks the Fundamental Research Funds for the Central Universities (ZYGX2015J102) and National Natural Science Foundation of China (71501025,11701070) for partial financial support. Financial support from the Swiss National Science Foundation Grant 200021-175752/1 is also kindly acknowledged.

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Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland
  2. 2.Department of Mathematics and CyberneticsWrocław University of EconomicsWrocławPoland
  3. 3.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduChina

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