A general linear birth and growth model

Abstract

We consider a birth and growth model where points (‘seeds’) arrive on a line randomly in time and space and proceed to ‘cover’ the line by growing at a uniform rate in both directions until an opposing branch is met; points which arrive on covered parts of the line do not contribute to the process. Existing results concerning the number of seeds assume that points arrive according to a Poisson process, homogeneous on the line, but possibly inhomogeneous in time. We derive results under less stringent assumptions, namely that the arrival process be a stationary simple point process.