Pickands–Piterbarg Constants for Self-Similar Gaussian Processes

• Autorzy: Dębicki Krzysztof , Tabiś Kamil

Identyfikatory

DOI

10.37190/0208-4147.40.2.6

• Języki publikacji: EN

Abstrakty

EN

For a centered self-similar Gaussian process {Y (t) : t ∈ [0;∞)} and R≥0 we analyze the asymptotic behavior of [formula], for suitably chosen γ> 0. Additionally, we find bounds for H^R_Y , R > 0, and a surprising relation between HY and the classical Pickands constants.

Słowa kluczowe

EN

Gaussian process; extremes; Pickands constant; Piterbarg constant

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2020
• Tom: Vol. 40, Fasc. 2
• Strony: 297--315
• Opis fizyczny: Bibliogr.28 poz.

Twórcy

autor

Dębicki Krzysztof

• Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

autor

Tabiś Kamil

• Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Bibliografia

• [1] R. Adler, An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, Inst. Math. Statist., Hayward, CA, 1990.
• [2] J. M. P. Albin and H. Choi, A new proof of an old result by Pickands, Electron. Comm. Probab. 15 (2010), 339-345.
• [3] L. Bai, K. Dębicki, E. Hashorva, and L. Luo, On generalised Piterbarg constants, Methodology Computing Appl. Probab. 20 (2018), 137-164.
• [4] T. Bojdecki, L. G. Gorostiza, and A. Talarczyk, Sub-fractional Brownian motion and its relation to occupation times, Statist. Probab. Lett. 69 (2004), 405-419.
• [5] K. Burnecki and Z. Michna, Simulation of Pickands constants, Probab. Math. Statist. 22 (2002), 193-199.
• [6] K. Dębicki, Ruin probability for Gaussian integrated processes, Stoch. Process. Appl. 98 (2002), 151-174.
• [7] K. Dębicki, S. Engelke, and E. Hashorva, Generalized Pickands constants and stationary maxstable processes, Extremes 20 (2017), 493-517.
• [8] K. Dębicki and E. Hashorva, On extremal index of max-stable processes, Probab. Math. Statist. 27 (2017), 299-317.
• [9] K. Dębicki and E. Hashorva, Approximation of supremum of max-stable stationary processes & Pickands constants, J. Appl. Probab. 33 (2020), 444-464.
• [10] K. Dębicki, E. Hashorva, L. Ji, and T. Rolski, Extremal behavior of hitting a cone by correlated Brownian motion with drift, Stoch. Process. Appl. 128 (2018), 4171-4206.
• [11] K. Dębicki, E. Hashorva, L. Ji, and K. Tabi´s, Extremes of vector-valued Gaussian processes: Exact asymptotics, Stoch. Process. Appl. 125 (2015), 4039-4065.
• [12] K. Dębicki, E. Hashorva, and P. Liu, Uniform tail approximation of homogeneous functionals of Gaussian fields, Adv. Appl. Probab. 49 (2017), 1037-1066.
• [13] K. Dębicki and M. Mandjes, Exact overflow asymptotics for queues with many Gaussian inputs, J. Appl. Probab. 40 (2003), 704-720.
• [14] K. Dębicki, Z. Michna, and T. Rolski, Simulation of the asymptotic constant in some fluid models, Stoch. Models 19 (2003), 407-423.
• [15] K. Dębicki and K. Tabi´s, Extremes of the time-average of stationary Gaussian processes, Stoch. Process. Appl. 121 (2011), 2049-2063.
• [16] A. B. Dieker and T. Mikosch, Exact simulation of Brown-Resnick random fields at a finite number of locations, Extremes 18 (2015), 301-314.
• [17] K. Dzhaparidze and H. Van Zanten, A series expansion of fractional Brownian motion, Probab. Theory Related Fields 130 (2004), 39-55.
• [18] E. Hashorva, S. Kobelkov, and V. I. Piterbarg, On maximum of Gaussian process with unique maximum point of its variance, arXiv:1901.09753 (2019).
• [19] Ch. Houdré and J. Villa, An example of infinite dimensional quasi-helix, in: Stochastic Models (Mexico City, 2002), Contemp. Math. 336, Amer. Math. Soc., 2003, 195-202.
• [20] D. G. Konstant and V. I. Piterbarg, Extreme values of the cyclostationary Gaussian random process, J. Appl. Probab. 30 (1993), 82-97.
• [21] J. Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104 (1962), 62-78.
• [22] P. Lei and D. Nualart, A decomposition of the bifractional Brownian motion and some applications, Statist. Probab. Lett. 79 (2009), 619-624.
• [23] W. Li and Q. Shao, Lower tail probabilities for Gaussian processes, Ann. Probab. 32 (2004), 216-242.
• [24] Z. Michna, Remarks on Pickands’ theorem, Probab. Math. Statist. 37 (2017), 373-393.
• [25] J. Pickands, Asymptotic properties of the maximum in a stationary Gaussian process, Trans. Amer. Math. Soc. 145 (1969), 75-86.
• [26] V. I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields, Transl. Math. Monogr. 148, Amer. Math. Soc., 2012.
• [27] V. I. Piterbarg and V. P. Prisiazhniuk, Asymptotic analysis of the probability of large excursions for a nonstationary Gaussian process, Teor. Veroyatnost. Mat. Statist. 18 (1978), 121-134 (in Russian).
• [28] R. A. Vitale, The Wills functional and Gaussian processes, Ann. Probab. 24 (1996), 2172-2178.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).