TABLE OF CONTENTS CFTP is not the only method of perfect simulation. Here we describe a different method, due to Fill. We begin by discussing a historical predecessor, Siegmund duality ( 4.1); we use this to explain Fill s method ( 4.2). We then describe a striking relationship recently introduced between Fill s method and CFTP ( 4.3), which shows the first is actually a conditioned version of the second.
We then turn to questions of efficiency whether CFTP can always be competitive with an idealized MCMC implementation which somehow just knows how long the burn-in period should be ( 4.4). Finally we consider the link between dom CFTP and geometric ergodicity ( 4.5), and briefly present yet another variant on CFTP, the Backwards-Forwards Algorithm ( 4.6), which has strong links to dom CFTP.
An important alternative to CFTP makes fuller use of the notion of time reversal, as in the dead-leaves example, and Section 3.1 on queues. We begin with a beautiful duality.
Theorem 33: (Siegmund duality) Suppose X is a process on [0, ?). When is there another process Y satisfying the following?
Answer: [86] Exactly when X is (suitably regular and) stochastically mono tone: x ? x ? implies
Proof: [Outline] Use Equation (7) to check monotonicity, and Fubini s Theorem to derive the Chapman-Kolmogorov equations.
Remark 34: If X is not stochastically monotone then Equation (7) will yield negative transition probabilities for
