A Class of Renewal Processes Driven by a Birth and Death Process
2002, v.8, Issue 1, 1-42
This paper deals with a generalization of the class of renewal processes with absolutely continuous life length distribution, obtained by allowing a random environment to modulate the stochastic intensity of the renewal process. The random environment is a birth and death process with a finite state space. The modulation is based on a set of deterministic failure rate functions, which are associated with the different environment states. Renewal processes in this environment (RPRE's) are constructed by using a certain Poisson embedding technique. The coupling method is the main tool in this paper, and it turns out to be particularly useful when the underlying deterministic failure rates are increasing or decreasing. For such processes, domination results and stochastic monotonicity properties are established. The existence of a stationary RPRE process is investigated by considering an embedded regenerative process, and asymptotics, rate results and versions of Blackwell's theorem are investigated by establishing exact couplings. Particular attention is paid to properties not present in the standard renewal theory but which are due to the introduction of a random environment. Asymptotic normality and some expansions of the generalized renewal function are also considered.
Keywords: renewal and point processes,random environments,coupling,Poisson embedding,stochastic intensity,asymptotics,stochastic domination,stochastic monotonicity