Some New Results in Regenerative Process Theory
P. W. Glynn
Technical Report, Department of Operations Research, Stanford University (1982)
In this report, we investigate various properties of discrete-time regenerative processes (Xn). We start, in Section 2, by defining the concept of a regenerative stochastic process. In Section 3, ergodic theory for weakly regenerative processes is investigated. We prove strong laws for partial sum processes, and show that there exists an ergodic measure, pi which captures all the steady-state information of the process (Xn). Section 4 is concerned with construction of a stationary process from the ergodic measure of Section 3. Total variation convergence of certain measures to pi is studied, in the regenerative setting, in Section 5. Furthermore, rates of convergence, in terms of the moments of the inter-regeneration times, are obtained. These ideas are applied in Section 6 where it is shown that all regenerative processes are strong mixing, and that regenerative processes are uniformly strong mixing (phi-mixing) under a simple sufficient condition. These results allow elementary proofs of some classical mixing results of Davydov (1973) for Markov chains, and improve the results in the sense that estimates for the mixing constants can be obtained. Section 7 studies the central limit theorem (CLT) for weakly regenerative processes; this result can be used to obtain a new CLT for Markov chains; see 20. We also relate this CLT to a CLT for the corresponding stationary process, constructed in Section 4. Finally, in Section 8, we investigate the splitting property of regneration times (see Jacobsen (1974) for the definition of a splitting time) for a special class of Markov chains.