In this paper, we study a discrete-time analogue of a Hawkes process, modelled as a Poisson autoregressive process whose parameters depend on the past of the trajectory. The model is characterized to allow these parameters to take negative values, modelling inhibitory dynamics. More precisely, the model is the stochastic process ( Xn)n≥0 with parameters a1, . . . , ap ∈ R, p ∈ N and λ > 0, such that for all n ≥ p, conditioned on X0, . . . , Xn-1, Xn is Poisson distributed with parameter

This process can be seen as a discrete time Hawkes process with inhibition with a memory of length p. We first provide a sufficient condition for stability in the general case which is the analog of a condition for continuous time Hawkes processes from [3]. We then focus on the case p = 3, extending the results derived for the p = 2 case in a previous work [4]. In particular, we show that the process may be stable even if one of the coefficients ai is much greater than one.