A Review of Regenerative Processes

The authors present an expository review of the theory of regenerative processes starting with the more traditional notions and then moving on to some of the more recent and modern developments. First, classical regenerative processes are defined and some of their properties are reviewed. This includes discussion of time averages, weak and total variation convergence, and stationary versions of processes. Through examples, limitations of the classical definition are illustrated, and a more general definition is introduced, where cycle lengths are independently and identically distributed, but the cycles themselves are permitted to be dependent. Then the connection between this definition and Harris-recurrent Markov chains is shown, and both discrete and continuous-time processes, which turn out to have one-dependent cycles are discussed. In general, these continuous-time processes also have one-dependent cycle lengths, and hence are not regenerative, even by the more general definition. Some one-dependent continuous-time processes turn out to have independently and identically distributed cycle lengths, when looked at appropriately. The authors briefly discuss some recent work on this as well as briefly illustrate the connection between regenerative processes and the method of renovation found in the Russian literature. Finally, the most general notion of regenerative, synchronous processes are discussed and a general tightness result is given.