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The excursions of a stationary Gaussian process outside a large two-dimensional region

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Robert Illsley
Robert Illsley*
Affiliation:
London Guildhall University
*
∗ Current address: 2 Marble Hill Gardens, Twickenham, Middlesex, TW1 3AX, UK. Email address: robert@illsleys.freeserve.co.uk
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Abstract

Let X(t) be a continuous two-dimensional stationary Gaussian process with mean zero, having a marginal density function p[x] and covariance matrix R(t). Let Δ = {∂L; L > 0} be a family of piecewise smooth boundaries of similar two-dimensional star-shaped regions ΓL. We show that, under two conditions on R(t), the asymptotic distribution of the duration of an excursion of X(t) outside ΓL, for large L, depends on the position of the maximum of p[x] on ∂L and on whether R′(0) is zero or not, should the maximum occur at a vertex. We obtain the asymptotic distributions of the duration of an excursion for each of the three cases that arise. We also generalise some results of Breitung (1994) on the asymptotic crossing rates of vector Gaussian processes.

Keywords

multivariate Gaussian processeshigh level crossingsexcursionsasymptotic distributions

MSC classification

Primary: 60G10: Stationary processes 60G15: Gaussian processes
Secondary: 60G17: Sample path properties
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 1 , March 2001 , pp. 141 - 159
Copyright
Copyright © Applied Probability Trust 2001 

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