Regenerative structure of Markov chains simulated via common random numbers (Q1064678)

When simulating a number, say m, of stochastic systems a standard strategy is to use a common sequence of random numbers to drive all systems. \textit{P. Heidelberger} and \textit{D. Iglehart} [Adv. Appl. Probab. 11, 804-819 (1979; Zbl 0434.60076)] show that this can increase efficiency of simulations in many cases provided the m-vector of stochastic processes is, regarded as a single process, a positive recurrent regenerative process. The author shows that, provided each of the m processes is an irreducible positive recurrent Markov chain on a countable space, then the m-vector of processes is itself positive recurrent when restricted to a suitable subset of m-space. This result is shown to be untrue in general when the individual spaces are not countable.