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Correlation and variability in birth processes

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Peter Donnelly ,
Thomas Kurtz  and
Paul Marjoram
Peter Donnelly*
Affiliation:
Queen Mary and Westfield College, University of London
Thomas Kurtz*
Affiliation:
University of Wisconsin-Madison
Paul Marjoram*
Affiliation:
Queen Mary and Westfield College, University of London,
*
∗∗ Postal address: School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, UK.
∗∗∗ Postal address: Departments of Mathematics and Statistics, University of Wisconsin-Madison, Madison, WI 53706, USA.
∗∗∗ Present address: Department of Mathematics, Monash University, Clayton, VIC 3168, Australia.
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Abstract

Faddy (1990) has conjectured that the variability of a pure birth process is increased, relative to the linear case, if the birth rates are convex and decreased if they are concave. We prove the conjecture by relating variability to the correlation structure of certain more informative versions of the process. A correlation inequality due to Harris (1977) is used to derive the necessary positive and negative correlation results.

Keywords

NON-LINEAR BIRTH RATESRELATIVE VARIATIONCOUPLINGCORRELATION INEQUALITIES

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 2 , June 1993 , pp. 275 - 284
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

P. Donnelly was supported in part by SERC Advanced Fellowship B/AF/1255

P. Marjoram by SERC grant GR/F/32561.

Supported by a Visiting Fellowship from the SERC (GR/G/27485).

References

Arratia, R. (1981) Limiting point processes for rescalings of coalescing and annihilating random walks on ℤ d . Ann. Prob. 9, 909–936.Google Scholar
Arratia, R. (1983) The motion of a tagged particle in the simple symmetric exclusion system on ℤ1 . Ann. Prob. 11, 362–373.Google Scholar
Ball, F. and Donnelly, P. (1987) Interparticle correlation in death processes with application to variability in compartmental models. Adv. Appl. Prob. 19, 755–766.Google Scholar
Faddy, M. J. (1990) On variation in birth processes. Adv. Appl. Prob. 22, 480–483.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
Harris, T. E. (1977) A correlation inequality for Markov processes in partially ordered state spaces. Ann. Prob. 5, 451–454.Google Scholar
Liggett, T. M. (1985) Interacting Particle Systems. Springer-Verlag, New York.CrossRefGoogle Scholar