ABSTRACT
In this chapter, the authors derive a few consequences and extensions of the theory developed in Chapters 1, 5, and 6. The first two sections contain estimates similar to laws of large numbers, applied to some simple jump Markov processes. The authors show that the optimal path arising in a variational problem corresponds, in some sense, to a local change of measure: when is following, its jumps in direction occur at about rate where is the maximizing in the definition of. They study the limits and rates of convergence associated with empirical distributions of finite-state Markov processes. The limits considered are on the order of a law of large numbers, or of Kurtz’s Theorem; they are not about rare events. The authors do not attempt to derive sharp bounds, but derive bounds on the rate of convergence that are uniform over a class of Markov processes.
