Restricting the Range: Applications

Some general theorems on Markov chains are proved in Sections 2 through 6, using the theory developed in Chapter 1. In Section 7, there are hints for dealing with the transient case. A summary can be found at the beginning of Chapter 1. The sections of this chapter are almost independent of one another. For Sections 2 through 6, continue in the setting of Section 1.5. Namely, I is a finite or countably infinite set, with the discrete topology; and Ī = I for finite I, while Ī = I U {φ} is the one-point compactification of I for infinite I. There is a standard stochastic semigroup P on I,for which each i is recurrent and communicates with each j. For a discussion, see Section 1.4. The process X on the probability triple (Ω∞, P i ) is Markov with stationary transitions P, starting state i, and smooth sample functions. Namely, the sample functions are quasiregular and have metrically perfect level sets with infinite Lebesgue measure. For finite J ⊂ I, the process X J is X watched only when in J. This process has J-valued right continuous sample functions, which visit each state on a set of times of infinite Lebesgue measure. Relative to P i , the process X J is Markov with stationary transitions P J , and generator Q J = P′ J (0). For a discussion, see Section 1.6. Recall that X J visits states ξJ,0, ξJ,1, ... with holding times τJ,0, τJ,1 .... Recall that µ J (t) is the time on the X J -scale corresponding to time t on the X-scale, while γ J (t) is the largest time on the X-scale corresponding to time t on the X J -scale.