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

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.