$I$-Projection and Conditional Limit Theorems for Discrete Parameter Markov Processes

Abstract

Let $(X, \mathscr{B})$ be a compact metric space with $\mathscr{B}$ the $\sigma$-field of Borel sets. Suppose this is the state space of a discrete parameter Markov process. Let $C$ be a closed convex set of probability measures on $X$. Known results on the asymptotic behavior of the probability that the empirical distributions $\hat{P}_n$ belong to $C$ and new results on the Markov process distribution of $\omega_0, \ldots, \omega_{n - 1}$ under the condition $\hat{P}_n \in C$ are obtained simultaneously through a large deviations estimate. In particular, the Markov process distribution under the condition $\hat{P}_n \in C$ is shown to have an asymptotic quasi-Markov property, generalizing a concept of Csiszar.