具有厚尾分布逗留时间的更新理论：典型与罕见
Renewal theory with fat tailed distributed sojourn times: typical versus rare
摘要：
逗留时间服从重尾幂律分布的更新过程在物理建模中经常遇到，因此对感兴趣的观测量的典型波动进行了详细的研究。为了描述罕见事件，大偏差理论的速率函数方法不成立，必须考虑新的工具。在这里，我们调查的更新次数，向前和向后的复发时间，占用时间，和跨越观察时间的时间间隔的大偏差。我们展示了非归一化密度如何描述这些罕见的波动，以及如何从这些极限定律中获得某些可观测量的矩。数值模拟表明，我们的结果显示了反正弦，Dynkin，Darling-Kac，L{é}vy和Lamperti定律的偏差。
英文摘要：
Renewal processes with heavy-tailed power law distributed sojourn times are commonly encountered in physical modelling and so typical fluctuations of observables of interest have been investigated in detail. To describe rare events the rate function approach from large deviation theory does not hold and new tools must be considered. Here we investigate the large deviations of the number of renewals, the forward and backward recurrence time, the occupation time, and the time interval straddling the observation time. We show how non-normalized densities describe these rare fluctuations, and how moments of certain observables are obtained from these limiting laws. Numerical simulations illustrate our results showing the deviations from arcsine, Dynkin, Darling-Kac, L{é}vy and Lamperti laws.
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