论文标题
非线性鹰队过程的棕榈空间方法
A Palm Space Approach to Non-Linear Hawkes Processes
论文作者
论文摘要
$ \ r $上的霍克斯进程是一个点过程，其在时间$ t $ $ t $之前的强度功能是其过去活动的功能。它由其激活函数$φ$及其内存功能$ h $定义。在本文中，霍克斯属性以与其点之间距离相关的非阴性序列的子空间的子空间表示。通过使用固定点过程与其棕榈测量之间的经典对应关系，我们将相应的棕榈测量的表征作为马尔可夫核的不变分布。我们证明，如果$φ$是连续的，并且其增长率最多是线性的，速率低于某个常数，则存在一个固定的霍克斯点过程。无限函数$φ$的文献的经典Lipschitz条件放松了。我们的证明依赖于耦合方法的组合，线性鹰队过程的单调性能以及棕榈分布的经典结果。从$ \ r _- $扮演的空状态开始对霍克斯过程的调查也是一个重要的作用。此时重新审视了霍克斯和奥克斯的线性案例。 如果内存函数$ h $是指数函数，则在弱条件下，表明存在独特的固定鹰点过程。在这种情况下，其手掌度量是根据一维Harris Ergodic Markov链的不变分布表示的。当激活函数是具有度$ {>} 1 $的多项式$φ$时，就不存在固定的霍克斯进程，如果霍克斯进程从空状态开始，则获得了其点积累的缩放结果。
A Hawkes process on $\R$ is a point process whose intensity function at time $t$ is a functional of its past activity before time $t$. It is defined by its activation function $Φ$ and its memory function $h$. In this paper, the Hawkes property is expressed as an operator on the sub-space of non-negative sequences associated to distances between its points. By using the classical correspondence between a stationary point process and its Palm measure, we establish a characterization of the corresponding Palm measure as an invariant distribution of a Markovian kernel. We prove that if $Φ$ is continuous and its growth rate is at most linear with a rate below some constant, then there exists a stationary Hawkes point process. The classical Lipschitz condition of the literature for an unbounded function $Φ$ is relaxed. Our proofs rely on a combination of coupling methods, monotonicity properties of linear Hawkes processes and classical results on Palm distributions. An investigation of the Hawkes process starting from the null measure, the empty state, on $\R_-$ plays also an important role. The linear case of Hawkes and Oakes is revisited at this occasion. If the memory function $h$ is an exponential function, under a weak condition it is shown that there exists a unique stationary Hawkes point process. In this case, its Palm measure is expressed in terms of the invariant distribution of a one-dimensional Harris ergodic Markov chain. When the activation function is a polynomial $Φ$ with degree ${>}1$, there does not exist a stationary Hawkes process and if the Hawkes process starts from the empty state, a scaling result for the accumulation of its points is obtained.
