Johnson-Mehl tessellations: asymptotics and inferences

Seeds are randomly scattered in ℝ^d according to a spatial-temporal point process. Each seed has its own potential germination time. Each seed that succeeds in germinating will be the centre of a growing spherical inhibited region that prohibits germination of any seed with later potential germination time. The radius of an inhibited region at time t after the germination of the seed at its centre is vt. The set of locations first reached by the growth of the inhibited region originated from x is called the cell of x. The space will be partitioned into cells and this space-filling structure is called the Johnson–Mehl tessellation. We show that the time until a large cube is totally inhibited has an extreme value distribution. In particular, for d = 1, we obtain the exact distribution of this time by transforming the original process to a Markov process. Moreover, we prove a central limit theorem for the number of germinations within [0, L)^d. Finally, the maximum likelihood estimation for v, a nonparametric estimation for the intensity measure and for its density, and the maximum likelihood estimation for the parameters of the intensity with known analytical form are proposed.