We consider a nonnegative discrete time and bounded horizon process X for which 0 is an absorbing state and extend it by a random variable that is independent of X. We find a sufficient condition for the resulting process to satisfy, after a canonical time rescaling, the hypothesis of the monotone case theorem. If X describes a secretary type search on a poset with one maximal element or if we consider X with no extension then this condition assumes an especially simple log-concavity type form.
"Monotone case for an extended process." Adv. in Appl. Probab. 46 (4) 1106 - 1125, December 2014. https://doi.org/10.1239/aap/1418396245