Regenerative Processes

Abstract

Abstract A regenerative process is a stochastic process with the property that after some (usually) random time, it starts over in the sense that, from the random time on, the process is stochastically equivalent to what it was at the beginning. In many applied fields such as telecommunications, finance, and actuarial science, important probability models turn out to be regenerative. Properties of regenerative processes and the mathematical methods used in their analysis are fundamental in the analysis and understanding of these models. For classical regenerative processes, cycles and cycle lengths are i.i.d. (independent and identically distributed). This assumption has been weakened over time. Now, processes satisfying the first paragraph above are regarded as regenerative, without any independent assumptions. This article introduces the classical case in the section ‘Classical Regenerative Processes’, presents their time‐average properties in the section ‘Time‐average Properties, Classical Case’, and briefly treats more advanced topics in the section ‘Existence of Limits; Combining Processes’.