Ordered thinnings of point processes and random measures

This is a study of thinnings of point processes and random measures on the real line that satisfy a weak law of large numbers. The thinning procedures have dependencies based on the order of the points or masses being thinned such that the thinned process is a composition of two random measures. It is shown that the thinned process (normalized by a certain function) converges in distribution if and only if the thinning process does. This result is used to characterize the convergence of thinned processes to infinitely divisible processes, such as a compound Poisson process, when the thinning is independent and nonhomogeneous, stationary, Markovian, or regenerative. Thinning by a sequence of independent identically distributed operations is also discussed. The results here contain Renyi's classical thinning theorem and many of its extensions.