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 [1976] applied the censoring technique to show that each recurrent Markov chain has a positive regular measure unique to multiplication by a scalar. Freedman [1983] 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.

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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 [1976] applied the censoring technique to show that each recurrent Markov chain has a positive regular measure unique to multiplication by a scalar. Freedman [1983] used the censoring technique to approximate countable Markov chains for the limiting behavior.
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The censoring technique