A Random Environment Generalization of Lorden's Renewal Inequality
2002, v.8, Issue 4, 637-649
The aim of this paper is to obtain a generalization of Lorden's renewal inequality for a class of renewal processes in random environments. These processes (called RPRE's) are generalizations of the classical renewal processes with absolutely continuous life length distribution, and are obtained by allowing a random environment to modulate the stochastic intensity. The first part of this paper presents a probabilistic proof of Lorden's classical renewal inequality. The main ideas from this proof are generalized for regenerative point processes and a certain Lorden type inequality is obtained. Finally, this inequality is applied to RPRE's and the aimed generalization of Lorden's inequality is obtained. It takes a particularly transparent form when the RPRE's considered are of DFR or IFR type.
Keywords: renewal processes,regenerative point processes,random environments,coupling,Poisson embedding,failure rates,stochastic intensity,DFR,IFR