On renewal theory for cluster processes

We prove several forms of renewal theorem tailored to renewal processes with marks and clusters. In particular, for an i.i.d. sequence $(\xi_i,X_i)_{i \geq 0}$, 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 $T_i = X_0+\cdots + X_i$, $i \geq 0$, and corresponding renewal cluster process $\xi(\cdot )=\sum_{i\geq0}\xi_i(\,\cdot -T_i)$. 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.