Fill's algorithm for absolutely continuous stochastically monotone kernels (Q1860592)

\textit{J. A. Fill} [Probab. Eng. Inf. Sci. 12, No. 3, 283-302 (1998; Zbl 0978.62013)] introduced a perfect sampling algorithm for finite-state stochastically monotone Markov chains. Fill's algorithm was extended by \textit{J. A. Fill}, \textit{M. Machida}, \textit{D. J. Murdoch} and \textit{J. S. Rosenthal} [Random Struct. Algorithms 17, No. 3/4, 290-316 (2000) and in: Monte Carlo methods. Fields Inst. Commun. 26, 37-52 (2000; Zbl 0966.65008)] to generic chains on general (continuous) state spaces (this algorithm is denoted hereafter as FMMR algorithm). The aim of the present paper is to continue the investigation of the FMMR algorithm for absolutely continuous stochastically monotone kernels, and to show the correctness of the FMMR algorithm in the new framework under a set of (three) regularity conditions. The considered regularity conditions are proved to relax the previously known sufficient hypotheses on the FMMR algorithm. Furthermore, the possible applicability of the FMMR algorithm for the quasi-monotone case is introduced and discussed.