We find a Lyapunov-type sufficient condition for discrete-time Markov chains on a countable state space including an absorbing set to almost surely reach this absorbing set and to asymptotically stabilize conditional on nonabsorption. This result is applied to Bienaymè-Galton-Watson-like branching processes in which the offspring distribution depends on the current population size. This yields a generalization of the Yaglom limit. The techniques used mainly rely on the spectral theory of linear operators on Banach spaces and especially on the notion of quasi-compact linear operator.
"Aysmptotic Behavior of Absorbing Markov Chains Conditional on Nonabsorption for Applications in Conservation Biology." Ann. Appl. Probab. 11 (1) 261 - 284, February 2001. https://doi.org/10.1214/aoap/998926993