On renewal theory for cluster processes

Abstract

We prove several forms of renewal theorem tailored to renewal processes with marks and clusters. In particular, for an i.i.d. sequence $({\xi }_i,X_i{)}_{i\ge 0}$, where ${\xi }_0$ denotes a finite point process on $\mathbb{R}$ and $X_0$ denotes a nonnegative random variable of finite mean, we consider the renewal sequence $T_i=X_0+\cdots +X_i$, $i\ge 0$, and corresponding renewal cluster process $\xi (\cdot )={\sum }_{i\ge 0}{\xi }_i(\cdot -T_i)$. Under mild assumptions on the distribution of $(\xi ,X)$, we show by coupling methods that the generalized versions of Blackwell's renewal theorem, key renewal theorem, extended renewal theorem and elementary renewal theorem still hold, even with dependence between ${\xi }_i$'s and $X_i$'s.