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Extremes of Gaussian processes with smooth random expectation and smooth random variance

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Abstract

Let ξ(t), t ∈ [0, T],T > 0, be a Gaussian stationary process with expectation 0 and variance 1, and let η(t) and μ(t) be other sufficiently smooth random processes independent of ξ(t). In this paper, we obtain an asymptotic exact result for P(sup t∈[0,T](η(t)ξ(t) + μ(t)) > u) as u→∞.

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References

  1. R.J. Adler, G. Samorodnitsky, and T. Gadrich, The expected number of level crossings for stationary, harmonizable, symmetric, stable processes, Ann. Appl. Probab., 3:553–575, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  2. J.-M. Azais and M. Wschebor, Level Sets and Extrema of Random Processes and Fields, 1st ed., Wiley, New York, 2009.

    Book  MATH  Google Scholar 

  3. A. Baddeley, I. Bárány, R. Schneider, and W. Weil, Stochastic Geometry: Lectures Given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13–18, 2004, Springer, Berlin, Heidelberg, 2007.

  4. X. Fernique, Regularité des Trajectoires des Fonctions Aléatoires Gaussiennes, Lect. Notes Math., Vol. 480, Springer, Berlin, Heidelberg, 1975, pp. 2–97.

  5. E. Hashorva, A.G. Pakes, and Q. Tang, Asymptotics of random contractions, Insurance Math. Econ., 47:405–414, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Hüsler, V. Piterbarg, and E. Rumyantseva, Extremes of Gaussian processes with a smooth random variance, Stochastic Processes Appl., 121(11):2592–2605, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  7. M.R. Leadbetter, G. Lindgren, and H. Rootzen, Extremes and Related Properties of Random Sequences and Processes, 1st ed., Springer Ser. Stat., Springer, 1983.

  8. V. Piterbarg, G. Popivoda, and S. Stamatović, Extremes of Gaussian processes with a smooth random trend, Filomat, to appear.

  9. V.I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields, 1st ed., Transl. Math. Monogr., Vol. 148, AMS, Providence, RI, 1996.

  10. V.I. Piterbarg, Extremes for processes in random environments, in Encyclopedia of Environmetrics, 2nd ed., John Wiley & Sons, Chichester, 2012, pp. 976–978.

  11. V.I. Piterbarg, Twenty Lectures About Gaussian Processes, 1st ed., Atlantic Financial Press, London, New York, 2015.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Moscow Lomonosov State University, Leninskiye Gory, 119991, Moscow, Russia

    Vladimir Piterbarg

  2. University of Montenegro, GeorgeWashington blvd, 81000, Podgorica, Montenegro

    Goran Popivoda & Siniša Stamatović

Authors
  1. Vladimir Piterbarg
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  2. Goran Popivoda
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  3. Siniša Stamatović
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Correspondence to Vladimir Piterbarg.

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Piterbarg, V., Popivoda, G. & Stamatović, S. Extremes of Gaussian processes with smooth random expectation and smooth random variance. Lith Math J 57, 128–141 (2017). https://doi.org/10.1007/s10986-017-9347-2

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  • DOI: https://doi.org/10.1007/s10986-017-9347-2

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