Strongly Ergodic Behavior for Non-Stationary Markov Processes

This paper considers ergodic behavior of those non-stationary Markov processes which can be represented by a sequence of stochastic kernels, $\{P_n(x, y)\}$, defined on a $\sigma$-finite measure space $(S, \mathscr{F}, \mu)$. In particular, the convergence of the superpositions, $P_1P_2P_3 \cdots P_n$, of these kernels is related to the convergence of their corresponding left eigenfunctions, $\psi_n$, where $\psi_n(y) = \int \psi_n(x)P_n(x, y)\mu(dx)$ and $\int \psi_n(y)\mu(dy) = 1$. It is then shown how these results can easily be extended to the general case where densities are not assumed.