Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T07:06:40.730Z Has data issue: false hasContentIssue false

On the Growth of the One-Dimensional Reverse Immunization Contact Processes

Published online by Cambridge University Press:  14 July 2016

A. Tzioufas*
Affiliation:
Heriot-Watt University
*
Postal address: Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We are concerned with the variation of the supercritical nearest-neighbours contact process such that first infection occurs at a lower rate; it is known that the process survives with positive probability. Regarding the rightmost infected of the process started from one site infected and conditioned to survive, we specify a sequence of space-time points at which its behaviour regenerates and, thus, obtain the corresponding strong law and central limit theorem. We also extend complete convergence in this case.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Bezuidenhout, C. E. and Grimmett, G. (1990). The critical contact process dies out. Ann. Prob. 18, 14621482.Google Scholar
[2] Durrett, R. (1980). On the growth of one-dimensional contact processes. Ann. Prob. 8, 890907.Google Scholar
[3] Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Prob. 12, 9991040.Google Scholar
[4] Durrett, R. (1988). Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks/Cole, Pacific Grove, CA.Google Scholar
[5] Durrett, R. (1995). Ten Lectures on Particle Systems (Lecture Notes Math. 1608). Springer, New York.Google Scholar
[6] Durrett, R. and Griffeath, D. (1983). Supercritical contact processes on {Z}. Ann. Prob. 11, 115.Google Scholar
[7] Durrett, R. Schinazi, R. B. (2000). Boundary modified contact processes. J. Theoret. Prob. 13, 575594.Google Scholar
[8] Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems (Lecture Notes Math. 724). Springer, Berlin.Google Scholar
[9] Harris, T. E. (1972). Nearest-neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9, 6689.Google Scholar
[10] Kuczek, T. (1989). The central limit theorem for the right edge of supercritical oriented percolation. Ann. Prob. 17, 13221332.Google Scholar
[11] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.Google Scholar
[12] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin.Google Scholar
[13] Stacey, A. (2003). Partial immunization processes. Ann. Appl. Prob. 13, 669690.Google Scholar