Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-09T01:34:18.219Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aronowich, M. and Adler, R. J. (1985). Behaviour of χ2 processes at extrema. Adv. Appl. Prob. 17, 280–297.CrossRefGoogle Scholar
Aronowich, M and Adler, R. J. (1986). Extrema and level crossings of χ2 processes. Adv. Appl. Prob. 18, 901–920.CrossRefGoogle Scholar
Belyaev, Yu K. (1968). On the number of exits across the boundary of a region by a vector stochastic process. Theory Prob. Appl. 13, 320–324.CrossRefGoogle Scholar
Belyaev, Yu K. and Nosko, V P. (1969). Characteristics of excursions above a high level for a Gaussian process and its envelope. Theory Prob. Appl. 14, 296–309.CrossRefGoogle Scholar
Berman, S.M. (1971) Excursions above high levels for stationary Gaussian processes. Pacific J. Math. 36, 63–79.CrossRefGoogle Scholar
Bolotin, V. V. (1969). Statistical Methods in Structural Mechanics. Holden-Day, San Francisco.Google Scholar
Breitung, K. (1988). Asymptotic crossing rates for stationary Gaussian vector processes. Stoch. Proc. Appl. 29, 195–207.CrossRefGoogle Scholar
Breitung, K. (1994). Asymptotic approximations for probability integrals. Lecture Notes in Mathematics 1592. Springer-Verlag.Google Scholar
Cramér, H and Leadbetter, M. R. (1965). The moments of the number of crossings of a level by a stationary normal process. Ann. Math. Statist. 36, 1656–1663.CrossRefGoogle Scholar
Cramér, H and Leadbetter, M R. (1967). Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
de Maré, J. (1980). Optimal prediction of catastrophes with applications to Gaussian processes. Ann. Prob. 8, 841–850.Google Scholar
Hasofer, A. M. (1974). The upcrossing rate of a class of stochastic processes. In Studies in Probability and Statistics (papers in honour of Pitman, E. J. G.). ed. Williams, E. J.. Jerusalem Academic Press, Jerusalem. pp. 153–160.Google Scholar
Hasofer, A. M. and Lind, N. C. (1974). Exact and invariant second-moment code format. J. Eng. Mech. Div. ASCE 100, 111–121.CrossRefGoogle Scholar
Illsley, R. G. (1992). The crossings of boundaries by vector Gaussian processes with applications to problems in reliability. Thesis. City University, London.Google Scholar
Ito, K. (1964). The expected number of zeros of continuous stationary Gaussian processes. J. Math. Kyoto Univ. 3-2, 207–216.Google Scholar
Lind, N. C. (1977). Formulation of probabalistic design. J. Eng. Mech. Div. ASCE 103, 273–284.CrossRefGoogle Scholar
Lindgren, G. (1980a). Model processes in nonlinear prediction with application to detection and alarm. Ann. Prob. 8, 775–792.CrossRefGoogle Scholar
Lindgren, G. (1980b). Extreme values and crossings for the χ2 process and other functions of multidimensional Gaussian processes, with reliability applications. Adv. Appl. Prob. 12, 746–774.Google Scholar
Marcus, M. B. (1977). Level crossings of a stationary process with absolutely continuous sample paths. Ann. Prob. 5, 52–71.CrossRefGoogle Scholar
Sharpe, K. (1978). Some properties of the crossings process generated by a stationary χ2 process. Adv. Appl. Prob. 10, 373–391.CrossRefGoogle Scholar
Slepian, D. (1962). The one-sided barrier problem for Gaussian noise. Bell Syst. Tech. J. 41, 463–501.CrossRefGoogle Scholar
Veneziano, D, Grigoriu, M. and Cornell, C. A. (1977). Vector-process models for system reliability. J. Eng. Mech. Div. ASCE 103, 441–460.CrossRefGoogle Scholar
Ylvisaker, N. D. (1966). On a theorem of Cramér and Leadbetter. Ann. Math. Statist. 37, 682–685.CrossRefGoogle Scholar