Saichev A, Sornette D
Department of Management, Technology and Economics, ETH Zurich, Scheuchzerstrasse 7, CH-8092 Zurich, Switzerland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Feb;87(2):022815. doi: 10.1103/PhysRevE.87.022815. Epub 2013 Feb 22.
We study the statistical properties of recurrence times in the self-excited Hawkes conditional Poisson process, the simplest extension of the Poisson process that takes into account how the past events influence the occurrence of future events. Specifically, we analyze the impact of the power law distribution of fertilities with exponent α, where the fertility of an event is the number of triggered events of first generation, on the probability distribution function (PDF) f(τ) of the recurrence times τ between successive events. The other input of the model is an exponential law quantifying the PDF of waiting times between an event and its first generation triggered events, whose characteristic time scale is taken as our time unit. At short-time scales, we discover two intermediate power law asymptotics, f(τ)τ(-(2-α)) for τ<<τ(c) and f(τ)τ(-α) for τ(c)<<τ<<1, where τ(c) is associated with the self-excited cascades of triggered events. For 1<<τ<<1/ν, we find a constant plateau f(τ)=/const, while at long times, 1/ν</τ, f(τ)=/~e(-ντ) has an exponential tail controlled by the arrival rate ν of exogenous events. These results demonstrate a novel mechanism for the generation of power laws in the distribution of recurrence times, which results from a power law distribution of fertilities in the presence of self-excitation and cascades of triggering.
我们研究了自激霍克斯条件泊松过程中重现时间的统计特性，这是泊松过程最简单的扩展，它考虑了过去事件如何影响未来事件的发生。具体而言，我们分析了指数为α的生育率幂律分布的影响，其中事件的生育率是第一代触发事件的数量，对连续事件之间重现时间τ的概率分布函数（PDF）f(τ)的影响。该模型的另一个输入是一个指数律，用于量化事件与其第一代触发事件之间等待时间的PDF，其特征时间尺度被用作我们的时间单位。在短时间尺度下，我们发现了两种中间幂律渐近形式，当τ<<τ(c)时，f(τ)τ^(-(2 - α))，当τ(c)<<τ<<1时，f(τ)τ^(-α)，其中τ(c)与触发事件的自激级联相关。对于1<<τ<<1/ν，我们发现一个常数平台f(τ)=/const，而在长时间情况下，1/ν</τ，f(τ)=/~e^(-ντ)具有由外源事件到达率ν控制的指数尾部。这些结果展示了一种在重现时间分布中产生幂律的新机制，这是由自激和触发级联存在时生育率的幂律分布导致的。
