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Brownian motion and a sharply curved boundary

Published online by Cambridge University Press:  01 July 2016

A. D. Barbour
A. D. Barbour*
Affiliation:
Gonville and Caius College, Cambridge
*
∗Postal address: Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, U.K.
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Abstract

Daniels (1974) reduced the problem of approximating the distribution of the maximum size of a closed epidemic to that of finding the distribution of max0≦t≦2 {W(t) – N1/2c(t)}, where c is a smooth function with a unique minimum of 0 at t = 1, and he derived an approximation to this distribution which he showed to be accurate to order N–1/4. In this paper, his approximation is shown to be accurate to order N–1/3, and a refined approximation is given which is accurate to order N–1/2 log N. The new approximation is still normal, and its accuracy is similar to that of the original approximation of a discrete process by the Wiener process.

Keywords

CLOSED EPIDEMICCURVED BOUNDARYBROWNIAN MOTION
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 4 , December 1981 , pp. 736 - 750
Copyright
Copyright © Applied Probability Trust 1981 

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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
Daniels, H. E. (1974) The maximum size of a closed epidemic. Adv. Appl. Prob. 6, 607–621.Google Scholar
Komlós, J., Major, P. and Tusnády, G. (1975) An approximation of partial sums of independent RV'-s and the sample DF. I. Z. Wahrscheinlichkeitsth. 32, 111–131.Google Scholar
Nagaev, A. V. and Startsev, A. N. (1970) The asymptotic analysis of a stochastic model of an epidemic. Theory Prob. Appl. 15, 98–107.Google Scholar
Smith, R. L. (1979) Limit theorems for the reliability of series-parallel load-sharing systems. Technical Report No. 424, Cornell University.Google Scholar