Journal of the
Korean Mathematical Society JKMS

Consider a supercritical Bellman-Harris process evolving from one particle. We superimpose on this process the additional structure of movement. A particle whose parent was at $x$ at its time of birth moves until it dies according to a given Markov process $X$ starting at $x$. The motions of different particles are assumed independent. In this paper we show that when the movement process $X$ is standard Brownian the proportion of particles with position $ \le \sq t \; b$ and age $\le a$ tends with probability 1 to $A(a)\Phi(b)$ where $A(\cdot)$ and $\Phi(\cdot)$ are the stable age distribution and standard normal distribution, respectively. We also extend this result to the case when the movement process is a Levy process.

Keywords: Bellman-Harris process, branching Brownian motion, branching Levy processes, law of large numbers

MSC numbers: 60J80, 60J30