Asymptotics of a Class of Markov Processes Which Are Not in General Irreducible

Let $\mathbf{\alpha}_n$ be a sequence of i.i.d. nondecreasing random maps on a subset $S$ of $\mathbb{R}^k$ into itself and let $X_0$ be a random variable with values in $S$ independent of the sequence $\mathbf{\alpha}_n$. Then $X_n \equiv \mathbf{\alpha}_n \cdots \mathbf{\alpha}_1X_0$ is a Markov process. Conditions for the existence of unique invariant probabilities are obtained for such Markov processes which are not in general irreducible, extending earlier results of Dubins and Freedman to multidimensional and noncompact state spaces. In addition, a functional central limit theorem is obtained. These yield new results in time series and economic models.