The Growth and Spread of the General Branching Random Walk

Abstract

A general (Crump-Mode-Jagers) spatial branching process is considered. The asymptotic behavior of the numbers present at time $t$ in sets of the form $\lbrack ta, \infty)$ is obtained. As a consequence it is shown that if $B_t$ is the position of the rightmost person at time $t, B_t/t$ converges to a constant, which can be obtained from the individual reproduction law, almost surely on the survival set of the process. This generalizes the known discrete-time results.