On renewal theory for cluster processes

TL;DR

This paper extends classical renewal theorems to complex cluster processes with marks, demonstrating their validity under mild dependence conditions using coupling methods.

Contribution

It introduces generalized renewal theorems applicable to marked and clustered renewal processes with dependence, broadening the scope of renewal theory.

Findings

Generalized Blackwell's renewal theorem established

Key renewal theorem extended to cluster processes

Renewal theorems hold under mild dependence assumptions

Abstract

We prove several forms of renewal theorem tailored to renewal processes with marks and clusters. In particular, for an i.i.d. sequence $(\xi_i,X_i)_{i \geq 0}$, 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.