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

Abstract

Let (Z(t) : t⩾0) be a supercritical age-dependent branching process and let { Y n } 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 Y n (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 E e i tZ 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 )= E e i tZ 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.