Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T04:39:37.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Sur une procédure de branchement déterministe et ses dérivées aléatoires

Part of: Special processes Markov processes Classical measure theory

Published online by Cambridge University Press:  14 July 2016

Thierry Huillet  and
Andrzej Kłopotowski
Thierry Huillet*
Affiliation:
LIMHP-CNRS, Villetaneuse
Andrzej Kłopotowski*
Affiliation:
Université Paris XIII, Villetaneuse
*
∗ Postal address: Université Paris-Nord, Institut Galileé, Avenue J.-B. Clément, 93430 Villetaneuse, France.
∗ Postal address: Université Paris-Nord, Institut Galileé, Avenue J.-B. Clément, 93430 Villetaneuse, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with the description of both a deterministic and stochastic branching procedure. The renewal equations for the deterministic branching population are first derived which allow for asymptotic results on the ‘number' and ‘generation' processes. A probabilistic version of these processes is then studied which presents some discrepancy with the standard Harris age-dependent branching processes.

Keywords

BRANCHING PROCESSAGE-DEPENDENCEHAUSDORFF DIMENSION

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 28A78: Hausdorff and packing measures 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 2 , June 1994 , pp. 333 - 347
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bartlett, M. S. (1946) Stochastic processes. Lecture Notes, University of North Carolina.Google Scholar
[2] Bienayme, I. J. (1845) De la loi de multiplication et de la durée des familles. Soc. Philomath. Paris Extraits Ser. 5, 37–39.Google Scholar
[3] Billingsley, P. (1967) Ergodic theory and information. Wiley, New York.Google Scholar
[4] Gantmacher, F. R. (1966) Théorie des matrices. Dunod, Paris.Google Scholar
[5] Harris, T. E. (1963) The theory of branching processes. Springer-Verlag, Berlin.Google Scholar
[6] Heyde, C. C. and Seneta, E. (1972) Studies in the history of probability and statistics. XXXI. The simple branching process, a turning point test and a fundamental inequality: a historical note on I. J. Bienaymé. Biometrika 59, 680–683.Google Scholar
[7] Horn, R. A. and Johnson, C. R. (1985) Matrix Analysis. Cambridge University Press.Google Scholar
[8] Karlin, S. (1968) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
[9] Leslie, P. H. (1945) On the use of matrices in certain population mathematics. Biometrika 33, 183–212.Google Scholar
[10] Leslie, P. H. (1948) Some further notes on the use of matrices in population mathematics. Biometrika 35, 213–245.Google Scholar
[11] Lewis, E. G. (1942) On the generation and growth of a population. Sankhya 6, 93–96.Google Scholar
[12] Ludwig, D. (1978) Stochastic Population Theories. Lecture Notes in Biomathematics 3, Springer-Verlag, Berlin.Google Scholar
[13] Mandelbrot, B. B. (1982) The Fractal Geometry of Nature. W. H. Freeman, San Francisco.Google Scholar
[14] Mode, C. J. (1971) Multitype Branching Processes. American Elsevier, New York.Google Scholar
[15] Pollard, J. H. (1966) On the use of the direct matrix product in analysing certain stochastic population models. Biometrika 53, 397–415.Google Scholar