On the Lower Bound of Large Deviation of Random Walks

Abstract

In this note, we prove for a large class of random walks on $R^n$ that $\lim \inf_{n\rightarrow\infty}(1/n)\log P_x(L_n(\omega, \cdot) \in N) \geq - I(\mu)$ where $L_n(\omega, \cdot)$ is the occupation measure, $N$ is a weak neighborhood of $\mu$ and $I(\mu)$ is the usual Donsker-Varadhan functional. This generalizes a previous theorem of the author where the state space is assumed to be compact.