The censored Markov chain, also called watched Markov chain, was first considered by Lévy [1951, 1952, 1958]. Since then, the censored Markov chains have been very useful in the study of Markov chains. Kemeny, Snell and Knapp  applied the censoring technique to show that each recurrent Markov chain has a positive regular measure unique to multiplication by a scalar. Freedman  used the censoring technique to approximate countable Markov chains for the limiting behavior and also for more general issues. The censoring technique has been successfully applied to block-structured Markov chains and Markov renewal processes. The Li's 2010 book: Constructive Computation in Stochastic Models with Applications: The RG-Factorizations is an excellent overview on this research direction.