Normal approximations for discrete-time occupancy processes

Abstract

We study 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. Bounds on the rate of convergence for a central limit theorem are obtained using Stein's method and moment inequalities on the deviation from an analogous deterministic model. As a consequence, our work also implies a uniform law of large numbers for a subclass of these processes.

Fast approximate simulation of finite long-range spin systems

“…Following the same arguments as [23, Proposition 9], using Lemmas A.1 and A.2 in place of [23,Lemma 7], we arrive at a bound on the second-order weak error, presented in Lemma A.4. For brevity, we introduce the notation…”

“…To enable the application of standard techniques, as in [4,23], we compare η(t) with an independent site approximation ω(t) with transition rates…”

“…Proof. The proof proceeds in a similar fashion to [23,Proposition 9], comparing η(t) to ρ(t), ρ(t) to ρ δ (t), and finally ρ δ (t) to η δ (t). First, from Lemma A.4,…”

Mean Field and Refined Mean Field Approximations for Heterogeneous Systems: It Works!

“…For instance, this method has been used to develop higher-order diffusion approximations [2,5]. It is used in [23] to develop a normal approximation of a heterogeneous discrete time population process. One of the key differences between our work and theirs is that the two aforementioned papers consider one-dimensional processes (i.e., the state of each object of the system is either 0 or 1), and the extension to more complex dynamics is not direct, at least from a computational point of view.…”