Limit Theorems for Reversible Markov Processes

Abstract

Consider a reversible (self-adjoint) Markov process with a discrete time parameter and stationary transition probability functions satisfying the Harris recurrence condition. $\mathbf{P}^{(n)}(x, S)$ denotes the $n$-step transition probability function from $x$ to the measurable set $S$ and $\pi$ is the sigma-finite stationary measure induced by the above hypotheses. Using both a functional analytic representation for reversible probabilities and probabilistic identities, various limits are considered for both general and discrete spaces. The principle result gives necessary and sufficient conditions for sets $A$ and $\mathbf{B}$ so that a reversible, aperiodic Markov process satisfies the strong ratio limit property $\lim_{m\rightarrow\infty} \mathbf{P}^{(n + k)} (\mu, \mathbf{A})/\mathbf{P}^{(n)} (\nu, \mathbf{B}) = \pi(\mathbf{A})/\pi(\mathbf{B}$ where $\mu$ and v are arbitrary probability distributions defined on the space and $k$ is any integer.