Valid Discretization via Renewal Theory

As discussed in Chapter 2, an important drawback of Raftery and Lewis’ (1992a, 1996) convergence control method is that the discretized version of the Markov chain is not a Markov chain itself, unless a stringent lumpability condition holds (see Kemeny and Snell, 1960). This somehow invalidates the binary control method, although it provides useful preliminary information on the required number of iterations. However, the discrete aspect of the criterion remains attractive for its intuitive flavour and, while the Duality Principle of Chapter 1 cannot be invoked in every setting, this chapter shows how renewal theory can be used to construct a theoretically valid discretization method for general Markov chains. We then consider some convergence control methods based on these discretized chains, even though the chains can be used in many alternative ways (see also Chapter 5).