Correlation bounds, mixing and m-dependence under random time-varying network distances with an application to Cox-processes

We will consider multivariate stochastic processes indexed either by vertices or pairs of a dynamic network. Under network, we understand network with fixed vertex set and an edge which changes randomly over time. assume that the spatial dependence-structure conditional on behaves in following way: Close (or vertices) are dependent, while dependence decreases conditionally distance increases. make this intuition mathematically precise considering three concepts based correlation, β-mixing time-varying β-coefficients independence. These allow proving weak-dependence results, for example, exponential inequality, might be independent interest. In order to demonstrate use these application, study asymptotics (for growing networks) goodness fit test interaction model Cox-type counting processes. This is then applied bike-sharing data.