We consider an extreme renewal process with no-mean heavy-tailed Pareto(II) inter-renewals and shape parameter $\alpha$ where $0\lt\alpha \leq 1$. Two steps are required to derive integral expressions for the analytic probability density functions (pdfs) of the fixed finite time $t$ excess, age, and total life, and require extensive computations. Step 1 creates and solves a Volterra integral equation of the second kind for the limiting pdf of a basic underlying regenerative process defined in the text, which is used for all three fixed finite time $t$ pdfs. Step 2 builds the aforementioned integral expressions based on the limiting pdf in the basic underlying regenerative process. The limiting pdfs of the fixed finite time $t$ pdfs as $t\rightarrow \infty$ do not exist. To reasonably observe the large $t$ pdfs in the extreme renewal process, we approximate them using the limiting pdfs having simple well-known formulas, in a companion renewal process where inter-renewals are right-truncated Pareto(II) variates with finite mean; this does not involve any computations. The distance between the approximating limiting pdfs and the analytic fixed finite time large $t$ pdfs is given by an $L_{1}$ metric taking values in $(0,1)$, where “near $0$” means “close” and “near $1$” means “far”.

我们考虑一个极端的更新过程，具有非均值重尾 Pareto(II) 相互更新和形状参数$\alpha$其中$0\lt\alpha \leq 1$。需要两个步骤来导出固定有限时间$t$剩余、年龄和总寿命的解析概率密度函数 (pdf) 的积分表达式，并且需要大量计算。步骤 1 创建并求解第二类 Volterra 积分方程，用于文本中定义的基本基础再生过程的极限 pdf，该方程用于所有三个固定有限时间 $t$ pdf。步骤2基于基础再生过程中的极限pdf构建上述积分表达式。固定有限时间的极限 pdf$t$ pdf $t\rightarrow \infty$不存在。为了合理地观察极端更新过程中的大$t$ pdf，我们使用具有简单众所周知公式的极限 pdf 来近似它们，在同伴更新过程中，其中间更新是具有有限均值的右截断 Pareto(II) 变量; 这不涉及任何计算。近似极限 pdf 与解析固定有限时间大$t$ pdf之间的距离由$L_{1}$度量给出，取值$(0,1)$，其中“near $0$ ”表示“接近” “near $1$ ”意味着“远”。