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 ttt in sets of the form [ta,∞)[ta,∞)\lbrack ta, \infty) is obtained. As a consequence it is shown that if BtBtB_t is the position of the rightmost person at time t,Bt/tt,Bt/tt, 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.