TABLE OF CONTENTS An early misconception about CFTP was that it could be applied only to monotonic Markov chains. We have already seen a mild counterexample: monotonicity is not particularly evident in the dead leaves model (though it can be forced into a monotonic framework using the notion of region of occlusion ). More general treatments use ideas of regeneration, which we now introduce.
We begin by summarizing the theory of Markov chain small sets ( 2.1), a theoretical discretization method which allows us to perform small-set CFTP for Markov chains on continuous state space ( 2.2). We then survey variations on this theme: slice sampling ( 2.3), the multi-shift sampler ( 2.4), catalytic CFTP ( 2.5), read-once CFTP ( 2.6). These variations are all part of the tool-set for successful application of CFTP in practice. We conclude with a brief discussions of some more technical complements to small-set CFTP ( 2.7).
Suppose we desire to construct a coupling between two random variables X, Y yielding a maximal positive chance of X= Y and otherwise not subject to any constraint. (This is related to the notion of convergence station naire, or parking convergence , from stochastic process theory.) Clearly this coupling is relevant to CFTP, where we aspire to coalescence!
Given two overlapping probability densities f and g, we can implement such a coupling (X, Y) as follows:
Compute ?= ?( f ? g)(x) d x.
