论文标题
关于集群过程的更新理论
On renewal theory for cluster processes
论文作者
论文摘要
我们证明了针对带有标记和集群的更新过程量身定制的几种续订定理。特别是对于I.I.D. sequence $(ξ_i,X_i)_{i \geq 0}$, where $ξ_0$ 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续订群集进程$ξ（\ cdot）= \ sum_ {i \ geq0}ξ_i（\，\ cdot -t_i）$。在对$（ξ，x）$分布的轻度假设下，我们通过耦合方法表明，Blackwell续订定理的广义版本，KEY RENEWAL定理，扩展了续订定理和基本续订定理，即使在$ $ eCed_i $'s和$ x_i $ $'s和$ x_i $ $'s之间仍然存在。
We prove several forms of renewal theorem tailored to renewal processes with marks and clusters. In particular, for an i.i.d. sequence $(ξ_i,X_i)_{i \geq 0}$, where $ξ_0$ 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 $ ξ(\cdot )=\sum_{i\geq0}ξ_i(\,\cdot -T_i)$. Under mild assumptions on the distribution of $(ξ,X)$, 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 $ξ_i$'s and $X_i$'s.
