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We consider a discrete-time version of a Hawkes process defined as a Poissonauto-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 $(X_n)_{n\ge0}$ withparameters $a_1,\ldots,a_p \in \mathbb{R}$, $p\in\mathbb{N}$ and $\lambda \ge0$, such that for all $n\ge p$, conditioned on $X_0,\ldots,X_{n-1}$, $X_n$ isPoisson distributed with parameter \[ \left(a_1 X_{n-1} + \cdots + a_p X_{n-p}+ \lambda \right)_+ \] We consider specifically the case $p = 2$, for which weare able to classify the asymptotic behavior of the process for the whole rangeof parameters, except for boundary cases. In particular, we show that theprocess remains stochastically bounded whenever the linear recurrence equation$x_n = a_1x_{n-1} + a_2x_{n-1} + \lambda$ remains bounded, but the converse isnot true. Relatedly, the criterion for stochastic boundedness is not symmetricin $a_1$ and $a_2$, in contrast to the case of non-negative parameters,illustrating the complex effects of inhibition.

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