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The maximum of a Gaussian process whose mean path has a maximum, with an application to the strength of bundles of fibres

Published online by Cambridge University Press:  01 July 2016

H. E. Daniels
H. E. Daniels*
Affiliation:
University of Cambridge
*
∗Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge, CB2 1SB, UK.
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Abstract

Daniels and Skyrme (1985) derived the joint distribution of the maximum, and the time at which it is attained, of a Brownian path superimposed on a parabolic curve near its maximum. In the present paper the results are extended to include Gaussian processes which behave locally like Brownian motion, or a process transformable to it, near the maximum of the mean path. This enables a wider class of practical problems to be dealt with. The results are used to obtain the asymptotic distribution of breaking load and extension of a bundle of fibres which can admit random slack or plastic yield, as suggested by Phoenix and Taylor (1973). Simulations confirm the approximations reasonably well. The method requires consideration not only of a Brownian bridge but also of an analogous process with covariance function t1(1 + t2), .

Keywords

LOCAL BROWNIAN MOTIONCURVED BOUNDARYEXTREME VALUES
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 2 , June 1989 , pp. 315 - 333
Copyright
Copyright © Applied Probability Trust 1989 

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References

Barbour, A. D. (1975) A note on the maximum size of a closed epidemic. J. R. Statist. Soc. B 37, 459–460.Google Scholar
Breiman, L. (1968) Probability. Addison-Wesley, Reading, Mass.Google Scholar
Chernoff, H. (1964) Estimation of the mode. Ann. Inst. Statist. Math. 16, 31–41.Google Scholar
Daniels, H. E. (1945) The statistical theory of the strength of bundles of threads. I. Proc. R. Soc. London A 183, 404–435.Google Scholar
Daniels, H. E. (1974) The maximum size of a closed epidemic. Adv. Appl. Prob. 6, 607–621.CrossRefGoogle Scholar
Daniels, H. E. and Skyrme, T. H. R. (1985) The maximum of a random walk whose mean path has a maximum. Adv. Appl. Prob. 17, 85–99.Google Scholar
Durbin, J. (1985) The first passage density of a continuous Gaussian process to a general boundary. J. Appl. Prob. 22, 99–122.Google Scholar
Groeneboom, P. (1984) Preprint.Google Scholar
Phoenix, S. L. and Taylor, H. M. (1973). The asymptotic strength of a general fiber bundle. Adv. Appl. Prob. 5, 200–216.Google Scholar
Smith, R. L. (1982). The asymptotic distribution of the strength of a series-parallel system with equal load sharing. Ann. Prob. 10, 137–171.Google Scholar