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

Abstract

We consider a discrete-time version of a Hawkes process defined as a Poisson auto-regressive process whose parameters depend on the past of the trajectory. We allow these parameters to take on 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 $λ\ge 0$, such that for all $n\ge p$, conditioned on $X_0,\ldots,X_{n-1}$, $X_n$ is Poisson distributed with parameter \[ \left(a_1 X_{n-1} + \cdots + a_p X_{n-p} + λ\right)_+ \] 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 linear recurrence equation $x_n = a_1x_{n-1} + a_2x_{n-1} + λ$ remains bounded, but the converse is not true. Relatedly, 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.

Figures (8)

• Figure 1: The partition of the parameter space described in Theorem \ref{['thm:classification']}. The green region corresponds to $\mathcal{R}$, while the red region corresponds to $\mathcal{T}$. The smaller figures are typical trajectories of the Markov chain $(X_n)_{n\ge0}$ for each region of the parameter space. In the all the simulations, we chose $\lambda=1$. The region delineated by a dashed triangular line corresponds to the region of parameter space for which the linear recurrence equation $y_{n} = ay_{n-1} + by_{n-2} + \lambda$ is bounded for all $n \in \mathbb{N}$, for any given $y_0, y_1 \in \mathbb{R}$. See Appendix A for more details.
• Figure 2: Illustration of the three zones of parameters on which the proof of ergodicity will be carried.
• Figure 3: An illustration of Case $\mathcal{R}_2$. Here, the parameters are $a = 3, b = -2.5$ and $N=1000$. In red, the set $A$ of couples $( i, j )$ such that $s_{ij}=s_{0i}=0$.
• Figure 4: Graphical representation of the sets $A,C$ described above.
• Figure 5: Log-log plot of a typical trajectory of $(X_n)$, to make the erratic behaviour of the first points of the Markov chain more visible. Here, the parameters are $a = -0.3, b=1.2$ and $N=100$.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Proposition 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof