A law of large numbers on randomly deleted sets

Abstract

Consider a system into which units having random magnitude enter at arbitrary times and remain "active" (present in the system) for random periods. Suppose units of high magnitude have stochastically greater lifetimes (tend to stay active for longer periods) than units of low magnitude. Of interest is the process ${\mu (t): t \geq 0}$ where $\mu (t)$ denotes the average magnitude of all units active at time t. We give conditions which guarantee the convergence of $\mu (t)$ and we determine the form of the limit. Some related processes are also studied.