Preservation properties of a renewal process stopped at a random dependent time

One of the interesting problems on the stochastic behavior of random recurrent events in a random time interval is to obtain the conditions under which the reliability properties of a random time T are inherited by N(T), where {N(t):t≥0} is a stochastic process. Most of the studies on the topic has been done under the assumption that the random time T and the stochastic process {N(t):t≥0} are stochastically independent. However, in practice, there can be different cases when appropriate dependence structure is more appropriate. In this paper, we study the preservation of a renewal process stopped at a random time when they are stochastically dependent. We discuss the stochastic ordering properties and the preservation of reliability classes for the random counting variables N(T) when the corresponding counting process is a renewal process. Furthermore, we study the preservation of NBUE (NWUE) reliability class when the counting process is a homogeneous Poisson process.