Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-18T08:41:59.293Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotics of fluctuations in Crump‒Mode‒Jagers processes: the lattice case

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  01 February 2019

Svante Janson
Svante Janson*
Affiliation:
Uppsala University
*
Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden.
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.

Consider a supercritical Crump‒Jagers process in which all births are at integer times (the lattice case). Let μ̂(z) be the generating function of the intensity of the offspring process, and consider the complex roots of μ̂(z)=1. The root of smallest absolute value is e-α=1∕m, where α>0 is the Malthusian parameter; let γ* be the root of second smallest absolute value. Subject to some technical conditions, the second-order fluctuations of the age distribution exhibit one of three types of behaviour: (i) when γ*>e-α∕2=m-1∕2, they are asymptotically normal; (ii) when γ*=e-α∕2, they are still asymptotically normal, but with a larger variance; and (iii) when γ*<e-α∕2, the fluctuations are in general oscillatory and (degenerate cases excluded) do not converge in distribution. This trichotomy is similar to what has been observed in related situations, such as some other branching processes and for Pólya urns. The results lead to a symbolic calculus describing the limits. The asymptotic results also apply to the total of other (random) characteristics of the population.

Keywords

Crump‒Mode‒Jagers processage distribution

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F05: Central limit and other weak theorems
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 141 - 171
Copyright
Copyright © Applied Probability Trust 2018 

References

[1]Asmussen, S. and Hering, H. (1983).Branching Processes.Birkhäuser,Boston, MA.Google Scholar
[2]Billingsley, P. (1968).Convergence of Probability Measures.John Wiley,New York.Google Scholar
[3]Bohr, H. (1925).Zur Theorie der fastperiodischen Funktionen, I.Acta Math. 45,29–127.Google Scholar
[4]Dunford, N. and Schwartz, J. T. (1958).Linear Operators. I. General Theory.Interscience,London.Google Scholar
[5]Duren, P. L. (1970).Theory of H p Spaces.Academic Press,New York.Google Scholar
[6]Hall, P. and Heyde, C. C. (1980).Martingale Limit Theory and Its Applications.Academic Press,New York.Google Scholar
[7]Jagers, P. (1975).Branching Processes with Biological Applications.John Wiley,London.Google Scholar
[8]Janson, S. (1997).Gaussian Hilbert Spaces.Cambridge University Press.Google Scholar
[9]Janson, S. (2004).Functional limit theorems for multitype branching processes and generalized Pólya urns.Stoch. Process. Appl. 110,177–245.Google Scholar
[10]Nerman, O. (1981).On the convergence of supercritical general (C-M-J) branching processes.Z. Wahrscheinlichkeitsth. 57,365–395.Google Scholar
[11]Pouyanne, N. (2008).An algebraic approach to Pólya processes.Ann. Inst. H. Poincaré Prob Statist. 44,293–323.Google Scholar