Large Deviation Lower Bounds for Additive Functionals of Markov Processes

Abstract

Let $X_1, X_2,\ldots$ be a Markov process with state space $E$, a Polish space. Let $L_n(\omega, A) = n^{-1}\sum^{n - 1}_{j = 0}1_A(X_j(\omega))$ denote the normalized occupation time measure. If $\mu$ is a probability measure on $E, G$ is a weak neighborhood of $\mu$, and if $V \subset E$, then we obtain asymptotic lower bounds for probabilities $P^x\lbrack L_n(\omega, \cdot) \in G, X_j(\omega) \in V, 0 \leq j \leq n - 1 \rbrack$ in terms of $I(\mu)$, the rate function of Donsker and Varadhan. Our assumptions are weaker than those imposed by Donsker and Varadhan, and the proof works without any essential change in the continuous time case as well. In fact, the same proofs apply to certain bounded additive functionals: Let $r \geq 0$ and let $f: \Omega \rightarrow \mathbf{B}$ be bounded $\mathscr{F}^0_r$-measurable, where $\Omega$ is the sample space with the product topology (Skorohod topology in the continuous time case) and $\mathbf{B}$ is a separable Banach space; let $\theta_k: \Omega \rightarrow \Omega$ be the shift operator, i.e., $\theta_k\omega(j) = \omega(k + j)$. Then we get lower bounds for probabilities involving $n^{-1}(f(\omega) + f(\theta_1\omega) + \cdots + f(\theta_{n - 1}\omega))$ in place of $L_n(\omega, \cdot)$. In this latter situation, the rate function has to be the entropy function $H(Q)$ of Donsker and Varadhan.