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Asymptotics Of Fluctuations In Crump-Mode-Jagers Processes: The Lattice Case
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2018 (English)In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 50, no A, p. 141-171Article in journal (Refereed) Published
Abstract [en]

Consider a supercritical Crump-Mode-Jagers process in which all births are at integer times (the lattice case). Let (mu) over cap (z) be the generating function of the intensity of the offspring process, and consider the complex roots of (mu) over cap (z) = 1. The root of smallest absolute value is e(-alpha) = 1/m, where alpha > 0 is the Malthusian parameter; let gamma* 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 gamma* > e(-alpha/2) = m(-1/2), they are asymptotically normal; (ii) when gamma* = e(-alpha/2), they are still asymptotically normal, but with a larger variance; and (iii) when gamma* < e(-alpha/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 Polya 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.

Place, publisher, year, edition, pages
2018. Vol. 50, no A, p. 141-171
Keywords [en]
Crump-Mode-Jagers process, age distribution
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:uu:diva-377288DOI: 10.1017/apr.2018.76ISI: 000457454600014OAI: oai:DiVA.org:uu-377288DiVA, id: diva2:1289559
Available from: 2019-02-18 Created: 2019-02-18 Last updated: 2019-02-18Bibliographically approved

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Janson, Svante

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