On Transience Conditions for Markov Chains

Let X={Xn} n≥ 0, X0= const, be a (generally) time-inhomogeneous Markov chain taking values in a measurable space (X, B). Let L: X→[0,∞) be an unbounded measurable function; ie, the complete preimage L− 1 ([c,∞)) of the set [c,∞) is nonempty for every c> 0. We study conditions that guarantee the as convergence of L (Xn) in n to infinity, ie, such conditions that the equation