Limit theorems for monotonic particle systems and sequential deposition

We prove spatial laws of large numbers and central limit theorems for the ultimate number of adsorbed particles in a large class of multidimensional random and cooperative sequential adsorption schemes on the lattice, and also for the Johnson–Mehl model of birth, linear growth and spatial exclusion in the continuum. The lattice result is also applicable to certain telecommunications networks. The proofs are based on a general law of large numbers and central limit theorem for sums of random variables determined by the restriction of a white noise process to large spatial regions.