Approximations and Limit Theorems for Discrete-Time Occupancy Processes

We study deterministic and normal approximations for a class of discrete-time occupancy processes, namely, Markov chains with transition kernels of product Bernoulli form. This class encompasses numerous models which appear in the complex networks literature, including stochastic patch occupancy models in ecology, network models in epidemiology, and a variety of dynamic random graph models. Moment inequalities on the deviation from an analogous deterministic model are presented, alongside bounds on the rate of convergence in law to a normal approximation under the Wasserstein and Kolmogorov metrics, which are obtained using Steinu0027s method. We are able to identify a subclass of weakly interacting processes that exhibit approximate Gaussian behaviour, and we develop a general strong law of large numbers and a central limit theorem for these processes. Applications to epidemic network models, metapopulation models, and convergence of dynamic random graphs, are discussed.