Asymptotics of fluctuations in Crump‒Mode‒Jagers processes: the lattice case. (1st February 2019)

• Record Type: Journal Article
• Authors: Janson, Svante
• Editors: Asmussen, S.; Klebaner, F.; Nerman, O.; Vatutin, V.
• Is Part Of: Advances in applied probability. Volume 50:Number A(2018)
• Page Start: 141
• Page End: 171
• Publication Date: 2019-02-01
• Subjects: Crump‒Mode‒Jagers process, -- age distribution; Primary 60J80, -- Secondary 60F05; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2018.76
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: 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.