In this article, the running average of a subordinator with a tempered stable distribution is considered. We investigate a family of previously unexplored infinite-activity subordinators induced by the probability distribution of the running average process and determine their jump intensity measures. Special cases including gamma processes and inverse Gaussian processes are discussed. Then we derive easily implementable formulas for the distribution functions, cumulants, and moments, as well as provide explicit estimates for their asymptotic behaviors. Numerical experiments are conducted for illustrating the applicability and efficiency of the proposed formulas. Two important extensions of the running average process and its equi-distributed subordinator are subsequently examined, with concrete applications to structural degradation modeling with memory and financial derivatives pricing in the presence of enhanced asymmetric leptokurtic feature, where their advantages relative to various existing models are highlighted together with the mention of Euler discretization and compound Poisson approximation techniques.

中文翻译：

---

具有应用程序的平均回火稳定从属

在本文中，考虑了具有回火稳定分布的从属子的运行平均值。我们调查了由运行平均过程的概率分布引起的一系列以前未探索的无限活动从属子，并确定了它们的跳跃强度测量值。讨论了包括伽马过程和逆高斯过程在内的特殊情况。然后我们推导出分布函数、累积量和矩的易于实现的公式，并为它们的渐近行为提供明确的估计。进行了数值实验以说明所提出公式的适用性和效率。随后检查了运行平均过程及其等分布从属程序的两个重要扩展，