(Almost) complete characterization of the stability of a discrete-time Hawkes process with inhibition and memory of length two

IF 0.7 4区 数学 Q3 STATISTICS & PROBABILITY
Manon Costa, Pascal Maillard, Anthony Muraro
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引用次数: 0

Abstract

We consider a Poisson autoregressive process whose parameters depend on the past of the trajectory. We allow these parameters to take negative values, modelling inhibition. More precisely, the model is the stochastic process $(X_n)_{n\ge0}$ with parameters $a_1,\ldots,a_p \in \mathbb{R}$ , $p\in\mathbb{N}$ , and $\lambda \ge 0$ , such that, for all $n\ge p$ , conditioned on $X_0,\ldots,X_{n-1}$ , $X_n$ is Poisson distributed with parameter $(a_1 X_{n-1} + \cdots + a_p X_{n-p} + \lambda)_+$ . This process can be regarded as a discrete-time Hawkes process with inhibition and a memory of length p. In this paper we initiate the study of necessary and sufficient conditions of stability for these processes, which seems to be a hard problem in general. We consider specifically the case $p = 2$ , for which we are able to classify the asymptotic behavior of the process for the whole range of parameters, except for boundary cases. In particular, we show that the process remains stochastically bounded whenever the solution to the linear recurrence equation $x_n = a_1x_{n-1} + a_2x_{n-2} + \lambda$ remains bounded, but the converse is not true. Furthermore, the criterion for stochastic boundedness is not symmetric in $a_1$ and $a_2$ , in contrast to the case of non-negative parameters, illustrating the complex effects of inhibition.
(具有抑制和长度为 2 的记忆的离散时间霍克斯过程稳定性的(几乎)完整表征
我们考虑一个泊松自回归过程,其参数取决于轨迹的过去。我们允许这些参数取负值,以模拟抑制作用。更准确地说,该模型是随机过程 $(X_n)_{n\ge0}$ ,参数为 $a_1,\ldots,a_p \in \mathbb{R}$ , $p\in\mathbb{N}$ , 和 $\lambda \ge 0$ 、这样,对于所有 $nge p$,以 $X_0,\ldots,X_{n-1}$为条件,$X_n$ 是泊松分布,参数为 $(a_1 X_{n-1} + \cdots + a_p X_{n-p} + \lambda)_+$ 。在本文中,我们开始研究这些过程的稳定性的必要和充分条件,这在一般情况下似乎是个难题。我们特别考虑了 $p = 2$ 的情况,对于这种情况,除了边界情况外,我们能够对整个参数范围内的过程渐近行为进行分类。我们特别指出,只要线性递推方程 $x_n = a_1x_{n-1} + a_2x_{n-2} + \lambda$ 的解保持有界,过程就保持随机有界,但反之则不然。此外,与非负参数的情况相反,随机有界性标准在 $a_1$ 和 $a_2$ 中并不对称,这说明了抑制作用的复杂影响。
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来源期刊
Journal of Applied Probability
Journal of Applied Probability 数学-统计学与概率论
CiteScore
1.50
自引率
10.00%
发文量
92
审稿时长
6-12 weeks
期刊介绍: Journal of Applied Probability is the oldest journal devoted to the publication of research in the field of applied probability. It is an international journal published by the Applied Probability Trust, and it serves as a companion publication to the Advances in Applied Probability. Its wide audience includes leading researchers across the entire spectrum of applied probability, including biosciences applications, operations research, telecommunications, computer science, engineering, epidemiology, financial mathematics, the physical and social sciences, and any field where stochastic modeling is used. A submission to Applied Probability represents a submission that may, at the Editor-in-Chief’s discretion, appear in either the Journal of Applied Probability or the Advances in Applied Probability. Typically, shorter papers appear in the Journal, with longer contributions appearing in the Advances.
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