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Brownian motion with quadratic killing and some implications

Published online by Cambridge University Press:  24 August 2016

Michael L. Wenocur
Michael L. Wenocur*
Affiliation:
Ford Aerospace and Communications Corporation
*
∗Postal address: 3939 Fabian Way, Palo Alto, CA 94303, USA.
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Abstract

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

Keywords

INTEGRATED SQUARED BROWNIAN MOTIONKAC FUNCTIONALKILLING TIMESWEIBULL-LIKE RELIABILITY DISTRIBUTION
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 4 , December 1986 , pp. 893 - 903
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

This paper is based in part on research supported by Army Research Office Contract DAAG29-82-K-0151 and by the Office of Naval Research Contract N00014–82-C-0620.

References

[1] Abramovitz, M. and Stegun, I. A. (1970) Handbook of Mathematical Functions. Dover, New York.Google Scholar
[2] Anderson, T. W. (1958) An Introduction to Multivariate Statistical Analysis. Wiley, New York.Google Scholar
[3] Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
[4] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[5] Cameron, R. H. and Martin, W. T. (1944) The Wiener measure of Hilbert neighborhoods in the space of real continuous functions. J. Math. Phys. 23, 195–209.CrossRefGoogle Scholar
[6] Chung, K. L. (1974) A Course in Probability Theory, 2nd edn. Academic Press, New York.Google Scholar
[7] Erdös, P. and Kac, M. (1946) On certain limit theorems of the theory of probability. Bull. Amer. Math. Soc. 52, 292–302.Google Scholar
[8] Ito, K. and Mckean, H. P. (1965) Diffusion Processes and Their Sample Paths. Springer-Verlag, Berlin.Google Scholar
[9] Johnson, N. L. and Kotz, S. (1970) Continuous Univariate Distributions — 2. Houghton Mifflin, Boston.Google Scholar
[10] Kac, M. (1946) On the average of a certain Wiener functional and a related limit theorem in calculus of probability Trans. Amer. Math. Soc. 59, 401–414.Google Scholar
[11] Kac, M. (1951) On some connections between probability theory and differential and integral equations. Proc. 2nd Berkeley Symp. Math. Statist. Prob. 189–215.Google Scholar
[12] Karlin, S. and Taylor, H. M. (1981) A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
[13] Lemoine, A. J. and Wenocur, M. L. (1985) On failure modeling. Naval Res. Logist. Quart. 32, 497–508.Google Scholar
[14] Rice, S. O. (1982) The integral of the absolute value of the pinned Wiener process — calculation of its probability density by numerical integration. Ann. Prob. 10, 240–243.CrossRefGoogle Scholar
[15] Rothman, E. D. and Woodroofe, M. (1972) A Cramer–Von-Mises type statistic for testing symmetry. Ann. Statist. 43, 2035–2038.Google Scholar
[16] Sansome, G. and Garretson, J. (1960) Lectures on the Theory of Functions of a Complex Variable. Vol. I. Noordhoff, Groningen.Google Scholar
[17] Shepp, L. A. (1966) Radon–Nikodym derivatives of Gaussian measures. Ann. Math. Statist. 37, 321–354.CrossRefGoogle Scholar
[18] Shepp, L. A. (1982) On the integral of the absolute value of the pinned Wiener process. Ann. Prob. 10, 234–239.CrossRefGoogle Scholar
[19] Yosida, K. (1960) Lectures on Differential and Integral Equations. Wiley-Interscience, New York.Google Scholar