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.
Citation
Robert J. Adler. Michael B. Marcus. Joel Zinn. "Central Limit Theorems for the Local Times of Certain Markov Processes and the Squares of Gaussian Processes." Ann. Probab. 18 (3) 1126 - 1140, July, 1990. https://doi.org/10.1214/aop/1176990738
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