Linking Discrete and Continuous Chains

Abstract

When comparing discrete and continuous Markov chains from a theoretical perspective (through, say, Kemeny and Snell, 1960, or Feller, 1970, vol. 1, for the former and Revuz, 1984, or Meyn and Tweedie, 1993, for the latter), a striking difference is the scale of the machinery needed to deal with continuous Markov chains and, as a corollary, the relative lack of intuitive basis behind theoretical results for continuous Markov chains. This gap is the major incentive for this book, in the sense that convergence controls methods must keep away both from the traps of ad hoc devices which are “seen” to work well on artificial and contrived examples, and from the quagmire of formal convergence results which, while being fascinating from a theoretical point of view, either fail to answer the true purpose of the analysis, i.e. to decide whether or not the chain(s) at hand have really converged, or require such an involved analysis that they are not customarily applicable besides case-study setups. This is also why techniques such as Raftery and Lewis (1996) are quite alluring, given their intuitive background and theoretical (quasi-)validity.