Constructing the General Markov Chain

In this chapter, I will construct the general Markov chain with continuous time parameter, countable state space, and stationary standard transitions. Let I be the countable state space. Let I n be a sequence of finite subsets of I which swell to I. For each n, suppose X n is Markov with continuous time parameter, has stationary standard transition, and has right continuous I n -valued step functions for sample functions. Suppose that X n is the restriction of X n+1 to I n , for all n. Here is the necessary and sufficient condition for the existence of a process X whose restriction to each I n is X n . The sum of the lengths of the I2, I3, … -intervals occurring before X1-time t is finite. If X is chosen with moderate care, it is automatically Markov with stationary standard transitions and smooth sample functions. Sections 1.5–6 show this construction is general. I will only do the work when I forms one recurrent class; but it is quite easy to drop this condition.