Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks

Quansheng Liu

Stochastic Processes and their Applications, 1999, vol. 82, issue 1, 61-87

Abstract: Let (Z(t): t[greater-or-equal, slanted]0) be a supercritical age-dependent branching process and let {Yn} be the natural martingale arising in a homogeneous branching random walk. Let Z be the almost sure limit of Z(t)/EZ(t)(t-->[infinity]) or that of Yn (n-->[infinity]). We study the following problems: (a) the absolute continuity of the distribution of Z and the regularity of the density function; (b) the decay rate (polynomial or exponential) of the left tail probability P(Z[less-than-or-equals, slant]x) as x-->0, and that of the characteristic function EeitZ and its derivative as t-->[infinity]; (c) the moments and decay rate (polynomial or exponential) of the right tail probability P(Z>x) as x-->[infinity], the analyticity of the characteristic function [phi](t)=EeitZ and its growth rate as an entire characteristic function. The results are established for non-trivial solutions of an associated functional equation, and are therefore also applicable for other limit variables arising in age-dependent branching processes and in homogeneous branching random walks.

Keywords: Age-dependent; branching; processes; Branching; random; walks; Martingales; Functional; equation; Absolute; continuity; Moments; of; negative; orders; Left; tails; Moments; Exponential; moments; Right; tail; Decay; rate; and; analiticity; of; characteristic; function; Growth; order; of; entire; characteristic; function (search for similar items in EconPapers)
Date: 1999
References: View complete reference list from CitEc
Citations: View citations in EconPapers (6)

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