Extended Renovation Theory and Limit Theorems for Stochastic Ordered Graphs ∗

We extend Borovkov’s renovation theory to obtain criteria for coupling-convergence of stochastic processes that do not necessarily obey stochastic recursions. The results are applied to an “infinite bin model”, a particular system that is an abstraction of a stochastic ordered graph, i.e., a graph on the integers that has (i, j), i < j, as an edge, with probability p, independently from edge to edge. A question of interest is an estimate of the length Ln of a longest path between two vertices at distance n. We give sharp bounds on C = limn→∞(Ln/n). This is done by first constructing the unique stationary version of the infinite bin model, using extended renovation theory. We also prove a functional law of large numbers and a functional central limit theorem for the infinite bin model. Finally, we discuss perfect simulation, in connection to extended renovation theory, and as a means for simulating the particular stochastic models considered in this paper.