Extremes of Gaussian chaos processes with trend

Bai, Long

doi:10.1016/j.jmaa.2019.01.026

Cited by 3 publications

(3 citation statements)

References 22 publications

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“…In the literature various results are known for supremum of functions of Gaussian vector processes, for instance for chi-square processes, chaos of Gaussian processes, order statistics of Gaussian processes, (see, e.g., Piterbarg, 1994Piterbarg, , 1996Hashorva and Ji, 2015;Bai, 2019) or reflected Gaussian processes modelling a queueing process with Gaussian input (see, e.g., Norros, 1994;Hüsler and Piterbarg, 1999;Dȩbicki, 2002;Piterbarg, 2001;Dȩbicki and Mandjes, 2003;Hüsler and Piterbarg, 2004;Dieker, 2005;Mandjes, 2007;Dȩbicki andLiu, 2016, 2018). In Section 3 we illustrate the applicability of Theorem 1.1 by the analysis of three diverse families of stochastic processes: 1)…”

Section: Introduction and First Resultsmentioning

confidence: 99%

Sojourn times of Gaussian and related random fields

Dȩbicki¹,

Hashorva²,

Liu³

et al. 2023

ALEA

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This paper is concerned with the asymptotic analysis of sojourn times of random fields with continuous sample paths. Under a very general framework we show that there is an interesting relationship between tail asymptotics of sojourn times and that of supremum. Moreover, we establish the uniform double-sum method to derive the tail asymptotics of sojourn times. In the literature, based on the pioneering research of S. Berman the sojourn times have been utilised to derive the tail asymptotics of supremum of Gaussian processes. In this paper we show that the opposite direction is even more fruitful, namely knowing the asymptotics of supremum of random processes and fields (in particular Gaussian) it is possible to establish the asymptotics of their sojourn times. We illustrate our findings considering i) two dimensional Gaussian random fields, ii) chi-process generated by stationary Gaussian processes and iii) stationary Gaussian queueing processes.

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Section: Introduction and First Resultsmentioning

confidence: 99%

Sojourn times of Gaussian and related random fields

Dȩbicki¹,

Hashorva²,

Liu³

et al. 2023

ALEA

View full text Add to dashboard Cite

show abstract

Section: Introduction and First Resultsmentioning

confidence: 99%

Sojourn times of Gaussian related random fields

Dȩbicki¹,

Hashorva²,

Liu³

et al. 2021

Preprint

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This paper is concerned with the asymptotic analysis of sojourn times of random fields with continuous sample paths. Under a very general framework we show that there is an interesting relationship between tail asymptotics of sojourn times and that of supremum. Moreover, we establish the uniform double-sum method to derive the tail asymptotics of sojourn times. In the literature, based on the pioneering research of S. Berman the sojourn times have been utilised to derive the tail asymptotics of supremum of Gaussian processes. In this paper we show that the opposite direction is even more fruitful, namely knowing the asymptotics of supremum o f random processes and fields (in particular Gaussian) it is possible to establish the asymptotics of their sojourn times. We illustrate our findings considering i) two dimensional Gaussian random fields, ii) chi-process generated by stationary Gaussian processes and iii) stationary Gaussian queueing processes.

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“…, n are locallystationary Gaussian processes, [8] obtains the extreme of the supremum of the locally-stationary Gaussian process. See [9,10] for more literature about locally stationary Gaussian processes.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Extremes of Locally-stationary Chi-square processes on discrete grids

Bai¹

2018

Preprint

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For X i (t), i = 1, . . . , n, t ∈ [0, T ] centered Gaussian processes, the chi-square process n i=1 X 2 i (t) appears naturally as limiting processes in various statistical models. In this paper, we are concerned with the exact tail asymptotics of the supremum taken over discrete grids of a class of locally stationary chi-square processes where X i (t), 1 ≤ i ≤ n are not identical. An important tool for establishing our results is a generalisation of Pickands lemma under the discrete scenario.An application related to the change-point problem is discussed.

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Extremes of Gaussian chaos processes with trend

Cited by 3 publications

References 22 publications

Sojourn times of Gaussian and related random fields

Sojourn times of Gaussian and related random fields

Sojourn times of Gaussian related random fields

Extremes of Locally-stationary Chi-square processes on discrete grids

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