Extremes of Gaussian processes with a smooth random variance
type: Item
instance of: scholarly article
title: Extremes of Gaussian processes with a smooth random variance
zbMATH Open document ID: 1228.60059
author: Ekaterina Rumyantseva
author: J. Hüsler
author: V. I. Piterbarg
published in: Stochastic Processes and their Applications
publication date: 2011-10-11T00:00:00Z
review text: Three main theorems are established in this paper. Let \((X(t))_{t\in\mathbb R}\) be a random process, whose conditional distribution, given the random element \(Y\), is Gaussian. The first main result provides the asymptotics of \(\text{P}(\sup_{t\in[0,T]}X(t)>u)\) as \(u\to\infty\). Let \((\xi(t))_{t\in\mathbb R}\) be a Gaussian process and set \(\eta(t):=\lambda-\zeta t^2/2\), \(t\in\mathbb R\), where \(\lambda\), \(\zeta\) are non-negative random variables being independent of \(\xi(\cdot)\). The second main result establishes the asymptotic behavior of \(\text{P}(\sup_{t\in[-T,T]}\xi(t)(\lambda-\xi t^2/2)>u)\) as \(u\to\infty\). The third main result provides the asymptotics of \(\text{P}(\sup_{t\in[0,T]}\xi(t)\eta(t)>u)\) as \(u\to\infty\) by extending the particular parabola process from Theorem 2 to smooth processes \(\eta(\cdot)\). Each result is, of course, established under appropriate conditions.
reviewed by: Michael Falk
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 5956357
zbMATH Keywords: Gaussian process
zbMATH Keywords: conditional Gaussian process
zbMATH Keywords: locally stationary
zbMATH Keywords: ruin probability
zbMATH Keywords: random variance
zbMATH Keywords: extremes
zbMATH Keywords: large deviations
zbMATH Keywords: fractional Brownian motion
MaRDI profile type: Publication
full work available at URL: j.spa.2011.06.006
OpenAlex ID: W2053403721
cites work: Q3935962
cites work: The expected number of level crossings for stationary, harmonisable, symmetric, stable processes
cites work: Analysis I. Integral presentations asymptotic methods
cites work: Q4096185
cites work: Extremes of Gaussian processes with random variance
cites work: Q5703636
cites work: Q4864754
cites work: Q3368970
cites work: Q3499639
DOI: 10.1016/J.SPA.2011.06.006
description: scientific article; zbMATH DE number 5956357