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High Excursions of a Quadratic form for a Gaussian Stationary Vector Process

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Abstract

Exact asymptotic behavior is given for high excursion probabilities of a quadratic form for a zero-mean Gaussian stationary vector process with Pickands’ type covariance matrix in the vicinity of zero. The case of a quadratic form with a positive maximum eigenvalue of order 1 is considered.

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References

  1. P. Albin, E. Hashorva, L. Ji, and C. Ling, Extremes and Limit Theorems for Difference of Chi-Type Processes, arXiv:1508.02758[math.PR] (2015).

  2. E. Hashorva, D. Korshunov, and V. Piterbarg, “Asymptotic expansion of Gaussian chaos via probabilistic approach,” Extremes, 18, No. 3, 315–347 (2015).

    Article  MathSciNet  Google Scholar 

  3. D. Korshunov, V. Piterbarg, and E. Hashorva, “On extremal behavior of Gaussian chaos,” Dokl. Math., 88, No. 2, 566–568 (2013).

    Article  MathSciNet  Google Scholar 

  4. D. Korshunov, V. Piterbarg, and E. Hashorva, “On the asymptotic Laplace method and its application to random chaos,” Math. Notes, 97, No. 5-6, 878–891 (2015).

    Article  MathSciNet  Google Scholar 

  5. V. I. Piterbarg, “High deviations for multidimensional stationary Gaussian process with independent components,” in: Stability Problems for Stochastic Models, VSP (1994), pp. 197–230.

  6. V. I. Piterbarg, “High excursions for nonstationary generalized chi-square processes,” Stoch. Process Appl., 51, 307–337 (1994).

    Article  MathSciNet  Google Scholar 

  7. V. I. Piterbarg, Asymptotic Methods in Theory of Gaussian Random Processes and Fields, Transl. Math. Monogr., Vol. 148, Amer. Math. Soc., Providence (1996).

  8. V. I. Piterbarg, “High extrema of Gaussian chaos processes,” Extremes, 19, No. 2, 253–272 (2016).

    Article  MathSciNet  Google Scholar 

  9. V. I. Piterbarg and S. Stamatović, “Limit theorem for high level a-upcrossings by χ2-process,” Probab. Theory Appl., 48, No. 4, 734–741, 2004.

    Article  MathSciNet  Google Scholar 

  10. V. I. Piterbarg and A. Zhdanov, “On probability of high extremes for product of two independent Gaussian stationary processes,” Extremes, 18, No. 1, 99–108 (2015).

    Article  MathSciNet  Google Scholar 

  11. A. I. Zhdanov, “Probabilities of high excursions for trajectories of product of two Gaussian stationary processes,” Probab. Theory Appl., 60, No. 3, 605–613 (2015).

    MathSciNet  Google Scholar 

  12. A. I. Zhdanov and V. I. Piterbarg, “High extremes of Gaussian chaos processes. A discrete time approximation approach,” Probab. Theory Appl., 63, No. 1, 3–28 (2018).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    A. I. Zhdanov

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  1. A. I. Zhdanov
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Correspondence to A. I. Zhdanov.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 23, No. 1, pp. 123–144, 2020.

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Zhdanov, A.I. High Excursions of a Quadratic form for a Gaussian Stationary Vector Process. J Math Sci 262, 476–492 (2022). https://doi.org/10.1007/s10958-022-05829-5

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  • DOI: https://doi.org/10.1007/s10958-022-05829-5

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