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Critical growth of a semi-linear process

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Ilya Molchanov ,
Vadim Shcherbakov  and
Sergei Zuyev
Ilya Molchanov*
Affiliation:
University of Berne
Vadim Shcherbakov*
Affiliation:
University of Glasgow
Sergei Zuyev*
Affiliation:
University of Strathclyde
*
∗ Postal address: Department of Mathematical Statistics and Actuarial Science, University of Berne, Sidlerstr. 5, CH-3012 Berne, Switzerland. Email address: ilya@stat.unibe.ch
∗∗ On leave from the Laboratory of Large Random Systems, Faculty of Mathematics and Mechanics, Moscow State University. Current address: CWI, PO Box 94079, 1090 GB Amsterdam, The Netherlands. Email address: v.shcherbakov@cwi.nl
∗∗∗ Postal address: Department of Statistics and Modelling Science, University of Strathclyde, Glasgow G1 1XH, UK. Email address: sergei@stams.strath.ac.uk
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Abstract

This paper is motivated by the modelling of leaching of bacteria through soil. A semi-linear process X t − may be used to describe the soil-drying process between rain showers. This is a backward recurrence time process that corresponds to the renewal process of instances of rain. If a bacterium moves according to another process h, then the fact that h(t) stays above X t − means that the bacterium never hits a dry patch of soil and so survives. We describe a critical behaviour of h that separates the cases when survival is possible with a positive probability from the cases when this probability vanishes. An explicit formula for the survival probability is obtained in case h is linear and rain showers follow a Poisson process.

Keywords

Borel-Cantelli lemmapoint processrenewal processsemi-linear process

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 355 - 367
Copyright
Copyright © Applied Probability Trust 2004 

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References

Chow, Y. S., and Robbins, H. (1961). On sums of independent random variables with infinite mean and ‘fair’ games. Proc. Nat. Acad. Sci. USA 47, 330–335.CrossRefGoogle Scholar
Cox, D. R. (1962). Renewal Theory. Methuen, London.Google Scholar
Cox, D. R., and Isham, V. (1988). A simple spatial–temporal model of rainfall. Proc. R. Soc. London A 415, 317–328.Google Scholar
Grandell, J. (1991). Aspects of Risk Theory. Springer, New York.CrossRefGoogle Scholar
Kemp, J. S. et al. (1992) Leaching of genetically modified pseudomonas fluorescens through organic soils: influence of temperature, soil pH, and roots. Biol. Fertility Soils 13, 218–224.CrossRefGoogle Scholar
Kesten, H. (1971). Sums of random variables with infinite expectation. Amer. Math. Monthly 78, 305–308.CrossRefGoogle Scholar
Loève, M. (1960). Probability Theory. Van Nostrand, New York.Google Scholar
Paterson, E. et al. (1993). Leaching of genetically modified pseudomonas fluorescens through organic soils: influence of soil type. Biol. Fertility Soils 15, 308–314.CrossRefGoogle Scholar