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, ₙ(x, y)\, defined on a σ-finite measure space (S, F, μ). In particular, the convergence of the superpositions, P₁P₂P₃ ⋯ Pₙ, of these kernels is related to the convergence of their corresponding left eigenfunctions, ψₙ, where ψₙ(y) = ∫ ψₙ(x)Pₙ(x, y)μ(dx) and ∫ ψₙ(y)μ(dy) = 1. It is then shown how these results can easily be extended to the general case where densities are not assumed.