Restricting the Range

Let X be a Markov chain with state space I,stationary standard transitions P,and smooth sample functions. For simplicity, suppose I forms one recurrent class of states relative to P. Let J be a finite subset of I. Let X _J be X watched only when in J; namely, X _J is obtained from X by ignoring the times t with X(t) ∉ J. Call X _J the restriction of X to J. This operation has been considered by Lévy (1951, 1952, 1958) and Williams (1966). The process X _J is defined formally in Section 5. The idea is to introduce γ _J (t),the rightmost time on the X-scale to the left of which X spends time t in J. Then X _J (t)= X[γ _J (t)]. Suppose J and K are finite subsets of I,with J ⊂ K. Then X _J = (X _K ) _J as shown in Section 5. The process X _J is Markov with stationary standard transitions, say P _J on J. This is proved in Section 6. The semigroups {P _J :J ⊂ I} are equicontinuous; consequently, X _J converges to X in probability and in q-lim† with probability 1 as J increases to I ; in particular, P _J converges to P as J increases to I. These results are proved in Section 7.