Central limit theorems for the local times of certain Markov processes and the squares of Gaussian processes (Q756285)

Let \(\{X_ t:\) \(t\geq 0\}\) be an \({\mathbb{R}}^ d\)-valued, symmetric right Markov process with stationary transition density, and let \(\hat X_ t\), \(t\geq 0\), denote the process killed at an exponential random time. A theorem of Dynkin relating the local time \(\{L_ x:\) \(x\in {\mathbb{R}}^ d\}\) of \(\hat X_ t\) and the Gaussian process \(\{G_ x:\) \(x\in {\mathbb{R}}^ d\}\) with covariance the Green's function of \(\hat X_ t\), is used to show that \(L_ x\) satisfies the central limit theorem in \(C({\mathbb{R}}^ d)\) if and only if G(x) is sample continuous. The proof requires a very nice lemma, namely that the square of a sample continuous Gaussian process satisfies the CLT on the space of continuous functions. In later work, \textit{M. Marcus} and \textit{J. Rosen} [Ann. Probab., to appear] obtain very interesting results for local times of symmetric Markov processes via Dynkin's isomorphism.