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The moments of the number of exits from a simply connected region

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Robert Illsley
Robert Illsley*
Affiliation:
London Guildhall University
*
∗ Postal address: 2 Marble Hill Gardens, Twickenham, Middlesex, TW1 3AX, UK.
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Abstract

We generalise the work of Cramér and Leadbetter, Ylvisaker and Ito on the level crossings of a stationary Gaussian process to multivariate processes. Necessary and sufficient conditions for the existence of the expected number of crossings E(C) of the boundary of a region of ℝp by a stationary vector stochastic process are obtained, along with an explicit formula for E(C) in the Gaussian case. A rigorous proof of Belyaev's integral formula for the factorial moments of the number of exits from a region of ℝp is given for a class of processes which includes Gaussian processes having a finite second order spectral moment matrix. Applications to χ2 processes are briefly considered.

Keywords

Crossingsexitslevel crossingsfactorial momentsvector processesGaussian processes

MSC classification

Primary: 60G15: Gaussian processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 1 , March 1998 , pp. 167 - 180
Copyright
Copyright © Applied Probability Trust 1998 

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