Powiadomienia systemowe
- Sesja wygasła!
Tytuł artykułu
Wybrane pełne teksty z tego czasopisma
Identyfikatory
Warianty tytułu
Języki publikacji
Abstrakty
This paper studies the supremum of chi-square processes with trend over a threshold-dependent-time horizon. Under the assumptions that the chi-square process is generated from a centered self-similar Gaussian process and the trend function is modeled by a polynomial function, we obtain the exact tail asymptotics of the supremum of the chi-square proces with trend. These results are of interest in applications in engineering, insurance, queueing and statistics, etc. Some possible extensions of our results are also discussed.
Czasopismo
Rocznik
Tom
Strony
1--20
Opis fizyczny
Bibliogr. 36 poz.
Twórcy
autor
- School of Mathematical Sciences and LPMC, Nankai University
- Department of Actuarial Science, University of Lausanne, Quartier UNIL-Dorigny, Bâtiment Extranef, CH-1015 Lausanne, Switzerland
autor
- Department of Actuarial Science, University of Lausanne, Quartier UNIL-Dorigny, Bâtiment Extranef, CH-1015 Lausanne, Switzerland
Bibliografia
- [1] R. J. Adler and J. E. Taylor, Random Fields and Geometry, Springer Monogr. Math., Springer, New York 2007.
- [2] J. M. P. Albin, On extremal theory for stationary processes, Ann. Probab. 18 (1) (1990), pp. 92-128.
- [3] S. Asmussen and H. Albrecher, Ruin Probabilities, second edition, Adv. Ser. Stat. Sci. Appl. Probab., Vol. 14, World Scientific, Hackensack, NJ, 2010.
- [4] S. M. Berman, Sojourns and extremes of stationary processes, Ann. Probab. 10 (1) (1982), pp. 1-46.
- [5] K. Dębicki, Ruin probability for Gaussian integrated processes, Stochastic Process. Appl. 98 (1) (2002), pp. 151-174.
- [6] K. Dębicki, E. Hashorva, and L. Ji, Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals, Extremes 17 (3) (2014), pp. 411-429.
- [7] K. Dębicki, E. Hashorva, and L. Ji, Gaussian risk models with financial constraints, Scand. Actuar. J. (6) (2015), pp. 469-481.
- [8] K. Dębicki, E. Hashorva, and L. Ji, Extremes of a class of non-homogeneous Gaussian random fields, Ann. Probab. 44 (2) (2016), pp. 984-1012.
- [9] K. Dębicki and M. Mandjes, Exact overflow asymptotics for queues with many Gaussian inputs, J. Appl. Probab. 40 (3) (2003), pp. 704-720.
- [10] K. Dębicki and G. Sikora, Finite time asymptotics of fluid and ruin models: multiplexed fractional Brownian motions case, Appl. Math. (Warsaw) 38 (1) (2011), pp. 107-116.
- [11] A. B. Dieker, Extremes of Gaussian processes over an infinite horizon, Stochastic Process. Appl. 115 (2) (2005), pp. 207-248.
- [12] A. B. Dieker and B. Yakir, On asymptotic constants in the theory of Gaussian processes, Bernoulli 20 (3) (2014), pp. 1600-1619.
- [13] L. Dümbgen and A. Wellner, Confidence bands for distribution functions: A new look at the law of the iterated logarithm, preprint, 2014.
- [14] P. Embrechts and M. Maejima, Selfsimilar Processes, Princeton Ser. Appl. Math., Princeton University Press, Princeton, NJ, 2002.
- [15] E. Hashorva and L. Ji, Approximation of passage times of γ-reflected processes with fBm input, J. Appl. Probab. 51 (3) (2014), pp. 713-726.
- [16] E. Hashorva and L. Ji, Piterbarg theorem for chi-processes with trend, Extremes 18 (1) (2015), pp. 37-64.
- [17] J. Hüsler and V. I. Piterbarg, Extremes of a certain class of Gaussian processes, Stochastic Process. Appl. 83 (2) (1999), pp. 257-271.
- [18] J. Hüsler and V. I. Piterbarg, A limit theorem for the time of ruin in a Gaussian ruin problem, Stochastic Process. Appl. 118 (11) (2008), pp. 2014-2021.
- [19] D. Jarušková, Detecting non-simultaneous changes in means of vectors, Test 24 (4) (2015).
- [20] D. Jarušková and V. I. Piterbarg, Log-likelihood ratio test for detecting transient change, Statist. Probab. Lett. 81 (5) (2011), pp. 552-559.
- [21] C. Klüppelberg and C. Kühn, Fractional Brownian motion as a weak limit of Poisson shot noise processes with applications to finance, Stochastic Process. Appl. 113 (2) (2004), pp. 333-351.
- [22] D. A. Korshunov, V. I. Piterbarg, and E. Hashorva On the extreme values of Gaussian chaos, Dokl. Akad. Nauk 452 (5) (2013), pp. 483-485.
- [23] S. Kuriki, Y. Harushima, H. Fujisawa, and N. Kurata, Approximate tail probabilities of the maximum of a chi-square field on multi-dimensional lattice points and their applications to detection of loci interactions, Ann. Inst. Statist. Math. 66 (2014), pp. 725-757.
- [24] M. R. Leadbetter, G. Lindgren, and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes, Springer, 1983.
- [25] G. Lindgren, Extreme values and crossings for the X2-process and other functions of multidimensional Gaussian processes, with reliability applications, Adv. in Appl. Probab. 12 (3) (1980), pp. 746-774.
- [26] P. Liu and L. Ji, Extremes of chi-square processes with trend, arXiv:1407.6501, 2014.
- [27] M. M. Meerschaert, W. Wang, and Y. Xiao, Fernique-type inequalities and moduli of continuity for anisotropic Gaussian random fields, Trans. Amer. Math. Soc. 365 (2013), pp. 1081-1107.
- [28] I. Norros, A storage model with self-similar input, Queueing Syst. 16 (1994), pp. 387-396.
- [29] A. G. Pakes, Convolution equivalence and infinite divisibility, J. Appl. Probab. 41 (2) (2004), pp. 407-424.
- [30] V. I. Piterbarg, High excursions for nonstationary generalized chi-square processes, Stochastic Process. Appl. 53 (2) (1994), pp. 307-337.
- [31] V. I. Piterbarg, Asymptotic methods in the theory of Gaussian processes and fields, Transl. Math. Monogr., Vol. 148, American Mathematical Society, Providence, RI, 1996.
- [32] V. I. Piterbarg and V. P. Prisjažnjuk, Asymptotic behavior of the probability of a large excursion for a nonstationary Gaussian process, Teor. Veroyatnost. Mat. Statist. 18 (1978), pp. 121-134.
- [33] C. Qualls and H. Watanabe, Asymptotic properties of Gaussian random fields, Trans. Amer. Math. Soc. 177 (1973), pp. 155-171.
- [34] T. Rolski, H. Schmidli, V. Schmidt, and J. L. Teugels, Stochastic Processes for Insurance and Finance, Wiley Ser. Probab. Stat., Wiley, Chichester 1999.
- [35] Z. Tan and E. Hashorva, Exact asymptotics and limit theorems for supremum of stationary X-processes over a random interval, Stochastic Process. Appl. 123 (8) (2013), pp. 2983-2998.
- [36] C. A. Tudor and Y. Xiao, Sample path properties of bifractional Brownian motion, Bernoulli 13 (4) (2007), pp. 1023-1057.
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-4952eed7-6657-4388-b9d4-baa3244fa331