Given i.i.d. point processes N1, N2,..., let the observations be p-thinnings N'1, N'2,..., where p is a function from the underlying space E (a compact metric space) to [0, 1], whose interpretation is that a point of Ni at x is retained with probability p(x) and deleted with probability 1-p(x)....

Laws of large numbers, central limit theorems, and laws of the iterated logarithm are obtained for discrete and continuous time Markov processes whose state space is a set of measures. These results apply to each measure-valued stochastic process itself and not simply to its real-valued functionals.