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.