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

Let (Z(t): t⩾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→∞) or that of Yn (n→∞). 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⩽x) as x→0, and that of the characteristic function EeitZ and its derivative as |t|→∞; (c) the moments and decay rate (polynomial or exponential) of the right tail probability P(Z>x) as x→∞, the analyticity of the characteristic function φ(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.