Recurrent Markov chains

Summary:Let $\left\{X_n,-\infty < n + \infty \right\}$ be a Markov process with stationary transition probabilities having a $\sigma$-finite stationary measure and satisfying a weak recurrence condition. We investigate the structure of the forward and backward tail $\sigma$-fields, $\Cal T_{+\infty}$ and $\Cal T_{-\infty}$, under a variety of situations. The main result is a representation theorem for the sets of $\Cal T_{+\infty}$; using this we develop a self-contained comprehensive treatment, deriving new as well as known theorems, including the decomposition into cyclically moving classes of processes satisfying the condition of Harris. The point of view and the techniques are probabilistic throughout