Title: Statistics of the number of renewals, occupation times and correlation in ordinary, equilibrium and aging alternating renewal processes Show Chinese title

Title: 普通、平衡和老化交替更新过程中的更新次数、占据时间和相关性的统计特性

Abstract: Renewal process is a point process where an inter-event time between successive renewals is an independent and identically distributed random variable. Alternating renewal process is a dichotomous process and a slight generalization of the renewal process, where the inter-event time distribution alternates between two distributions. We investigate statistical properties of the number of renewals and occupation times for one of the two states in alternating renewal processes. When both means of the inter-event times are finite, the alternating renewal process can reach an equilibrium. On the other hand, an alternating renewal process shows aging when one of the means diverges. We provide analytical calculations for the moments of the number of renewals, occupation time statistics, and the correlation function for several case studies in the inter-event-time distributions. We show anomalous fluctuations for the number of renewals and occupation times when the second moment of inter-event time diverges. When the mean inter-event time diverges, distributional limit theorems for the number of events and occupation times are shown analytically. These are known as the Mittag-Leffler distribution and the generalized arcsine law in probability theory. Show Chinese abstract

Abstract: 更新过程是一个点过程，其中连续更新之间的事件时间是一个独立同分布的随机变量。 交替更新过程是一种二元过程，是更新过程的一个轻微推广，其中事件之间的时间分布在两个分布之间交替。 我们研究了交替更新过程中两个状态之一的更新次数和占用时间的统计特性。 当事件之间的时间的两个均值都是有限时，交替更新过程可以达到平衡。 另一方面，当其中一个均值发散时，交替更新过程表现出老化现象。 我们提供了几种事件时间分布情况下的更新次数、占用时间统计和相关函数的矩的解析计算。 当事件之间的时间的二阶矩发散时，我们展示了更新次数和占用时间的异常波动。 当事件之间的时间的均值发散时，我们分析地展示了事件数量和占用时间的分布极限定理。 这些被称为概率论中的Mittag-Leffler分布和广义反正弦定律。