Mixed ratio limit theorems for Markov processes

LetP be a conservative and ergodic Markov operator onL ₁(X, Σ,m). We give a sufficient condition for the existence of a decompositionA _f ↑X such that for 0≦f, g ∈L _∞ (A _j ) and any two probability measuresμ andν weaker thanm , whereλ is theσ-finite invariant measure (which necessarily exists). Processes recurrent in the sense of Harris are shown to have this decomposition, and an analytic proof of the convergence of is deduced for such processes. This paper is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the direction of Professor S. R. Foguel, to whom the author is grateful for his helpful advice and kind encouragement.