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Extrema of multi-dimensional Gaussian processes over random intervals

Part of: Stochastic processes

Published online by Cambridge University Press:  28 February 2022

Lanpeng Ji  and
Xiaofan Peng Open the ORCID record for Xiaofan Peng [Opens in a new window]
Lanpeng Ji*
Affiliation:
University of Leeds
Xiaofan Peng*
Affiliation:
University of Electronic Science and Technology of China
*
*Postal address: School of Mathematics, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK.
**Postal address: School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China. Corresponding author’s email address: xfpeng@uestc.edu.cn
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Abstract

This paper studies the joint tail asymptotics of extrema of the multi-dimensional Gaussian process over random intervals defined as $P(u)\;:\!=\; \mathbb{P}\{\cap_{i=1}^n (\sup_{t\in[0,\mathcal{T}_i]} ( X_{i}(t) +c_i t )>a_i u )\}$ , $u\rightarrow\infty$ , where $X_i(t)$ , $t\ge0$ , $i=1,2,\ldots,n$ , are independent centered Gaussian processes with stationary increments, $\boldsymbol{\mathcal{T}}=(\mathcal{T}_1, \ldots, \mathcal{T}_n)$ is a regularly varying random vector with positive components, which is independent of the Gaussian processes, and $c_i\in \mathbb{R}$ , $a_i>0$ , $i=1,2,\ldots,n$ . Our result shows that the structure of the asymptotics of P(u) is determined by the signs of the drifts $c_i$ . We also discuss a relevant multi-dimensional regenerative model and derive the corresponding ruin probability.

Keywords

Joint tail asymptoticGaussian processperturbed random walkruin probabilityfluid modelfractional Brownian motionregenerative model

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 1 , March 2022 , pp. 81 - 104
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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