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The maximum of a random walk whose mean path has a maximum

Published online by Cambridge University Press:  01 July 2016

H. E. Daniels  and
T. H. R. Skyrme
H. E. Daniels*
Affiliation:
University of Cambridge
T. H. R. Skyrme*
Affiliation:
University of Birmingham
*
∗Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, UK.
∗∗ Postal address: Department of Mathematics, P.O. Box 363, University of Birmingham, Birmingham B15 2TT, UK.
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Abstract

This paper discusses the joint distribution of the maximum and the time at which it is attained, of a random walk whose mean path is a curvilinear trend which itself has a maximum. A typical example of such a problem is the distribution of the maximum number of infectives present during the course of an epidemic. Another example where the random walk is constrained to terminate at 0 after a given time is provided by the distribution of the strength and breaking extension of a bundle of fibres.

A diffusion approximation to the joint distribution is obtained for the general case of a Brownian bridge. In the commonest class of cases which includes the two examples mentioned, a certain integral equation has to be solved. Its solution enables the marginal distribution of the time to reach the maximum to be tabulated, and the marginal distribution of the maximum confirms the results previously obtained by Daniels (1974) and Barbour (1975). Of particular interest is the conditional expectation of the maximum for a given time of attainment which behaves asymmetrically.

Keywords

EXTREME VALUESBROWNIAN BRIDGEEPIDEMICBUNDLECURVED BOUNDARY
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 1 , March 1985 , pp. 85 - 99
Copyright
Copyright © Applied Probability Trust 1985 

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References

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