Extremes of vector-valued Gaussian processes with Trend

Long Bai; Krzysztof Dȩbicki; Peng Liu

doi:10.1016/j.jmaa.2018.04.069

Extremes of vector-valued Gaussian processes with Trend

Long Bai, Krzysztof Dȩbicki, Peng Liu^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

Let X(t)=(X₁(t),…,X_n(t)),t∈T⊂R be a centered vector-valued Gaussian process with independent components and continuous trajectories, and h(t)=(h₁(t),…,h_n(t)),t∈T be a vector-valued continuous function. We investigate the asymptotics of P{supt∈T⁡min1≤i≤n⁡(X_i(t)+h_i(t))>u} as u→∞. As an illustration to the derived results we analyze two important classes of X(t): with locally-stationary structure and with varying variances of the coordinates, and calculate exact asymptotics of simultaneous ruin probability and ruin time in a Gaussian risk model.

Original language	English
Pages (from-to)	47-74
Number of pages	28
Journal	Journal of Mathematical Analysis and Applications
Volume	465
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2018.04.069
Publication status	Published - 1 Sept 2018
Externally published	Yes

Keywords

Conjunction
Extremes
Pickands constant
Piterbarg constant
Vector-valued Gaussian process

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10.1016/j.jmaa.2018.04.069

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@article{163bb11cc6434d0d9b465ad50716e057,

title = "Extremes of vector-valued Gaussian processes with Trend",

abstract = "Let X(t)=(X1(t),…,Xn(t)),t∈T⊂R be a centered vector-valued Gaussian process with independent components and continuous trajectories, and h(t)=(h1(t),…,hn(t)),t∈T be a vector-valued continuous function. We investigate the asymptotics of P{supt∈T⁡min1≤i≤n⁡(Xi(t)+hi(t))>u} as u→∞. As an illustration to the derived results we analyze two important classes of X(t): with locally-stationary structure and with varying variances of the coordinates, and calculate exact asymptotics of simultaneous ruin probability and ruin time in a Gaussian risk model.",

keywords = "Conjunction, Extremes, Pickands constant, Piterbarg constant, Vector-valued Gaussian process",

author = "Long Bai and Krzysztof Dȩbicki and Peng Liu",

note = "Funding Information: We thank Enkelejd Hashorva for discussions and comments that improved presentation of the results of this contribution. Thanks to the Swiss National Science Foundation Grant 200021-175752/1 , whereas K. Dȩbicki also acknowledges partial support from NCN Grant No. 2015/17/B/ST1/01102 (2016–2019). Publisher Copyright: {\textcopyright} 2018 Elsevier Inc.",

year = "2018",

month = sep,

day = "1",

doi = "10.1016/j.jmaa.2018.04.069",

language = "English",

volume = "465",

pages = "47--74",

journal = "Journal of Mathematical Analysis and Applications",

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TY - JOUR

T1 - Extremes of vector-valued Gaussian processes with Trend

AU - Bai, Long

AU - Dȩbicki, Krzysztof

AU - Liu, Peng

N1 - Funding Information: We thank Enkelejd Hashorva for discussions and comments that improved presentation of the results of this contribution. Thanks to the Swiss National Science Foundation Grant 200021-175752/1 , whereas K. Dȩbicki also acknowledges partial support from NCN Grant No. 2015/17/B/ST1/01102 (2016–2019). Publisher Copyright: © 2018 Elsevier Inc.

PY - 2018/9/1

Y1 - 2018/9/1

N2 - Let X(t)=(X1(t),…,Xn(t)),t∈T⊂R be a centered vector-valued Gaussian process with independent components and continuous trajectories, and h(t)=(h1(t),…,hn(t)),t∈T be a vector-valued continuous function. We investigate the asymptotics of P{supt∈T⁡min1≤i≤n⁡(Xi(t)+hi(t))>u} as u→∞. As an illustration to the derived results we analyze two important classes of X(t): with locally-stationary structure and with varying variances of the coordinates, and calculate exact asymptotics of simultaneous ruin probability and ruin time in a Gaussian risk model.

AB - Let X(t)=(X1(t),…,Xn(t)),t∈T⊂R be a centered vector-valued Gaussian process with independent components and continuous trajectories, and h(t)=(h1(t),…,hn(t)),t∈T be a vector-valued continuous function. We investigate the asymptotics of P{supt∈T⁡min1≤i≤n⁡(Xi(t)+hi(t))>u} as u→∞. As an illustration to the derived results we analyze two important classes of X(t): with locally-stationary structure and with varying variances of the coordinates, and calculate exact asymptotics of simultaneous ruin probability and ruin time in a Gaussian risk model.

KW - Conjunction

KW - Extremes

KW - Pickands constant

KW - Piterbarg constant

KW - Vector-valued Gaussian process

UR - http://www.scopus.com/inward/record.url?scp=85046687020&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2018.04.069

DO - 10.1016/j.jmaa.2018.04.069

M3 - Article

AN - SCOPUS:85046687020

SN - 0022-247X

VL - 465

SP - 47

EP - 74

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -

Extremes of vector-valued Gaussian processes with Trend

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Keywords

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