Central Limit Theorems for the Local Times of Certain Markov Processes and the Squares of Gaussian Processes

Abstract

Let $\{X_t, t \geq 0\}$ be an $R^d$-valued, symmetric, right Markov process with stationary transition density. Let $\{\hat{X}_t, t \geq 0\}$ denote the version of $X_t$ "killed" at an exponential random time, independent of $X_t$. Associated with $\hat{X}_t$ is a Green's function $g(x, y)$, which we assume satisfies $0 < g(x, x) < \infty$ for all $x$ and a local time $\{L_x, x \in R^d\}$. It follows from an isomorphism theorem of Dynkin that $L_x$ has continuous sample paths whenever $\{G(x), x \in R^d\}$, a Gaussian process with covariance $g(x, y)$, does. In this paper we use Dynkin's theorem to show that $L_x$ satisfies the central limit theorem in the space of continuous functions on $R^d$ if and only if $G(x)$ has continuous sample paths. This result strengthens a result of Adler and Epstein on the construction of the free field by means of a central limit theorem involving the local time, in the case when the local time is a point indexed process. In order to apply Dynkin's theorem the following result is obtained: The square of a continuous Gaussian process satisfies the central limit theorem in the space of continuous functions.