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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) ₍₀ with parameters a₁, , aₚ, p and > 0, such that for all n p, conditioned on X₀, , X₍-₁, Xₙ is Poisson distributed with parameter \ (a₁ X₍-₁ + + aₚ X₍- +) _+. \ This process can be seen as a discrete time Hawkes process with inhibition with a memory of length p. %This work is an extension of a prior work where we studied the specific case p = 2, for which we were able to classify the asymptotic behaviour of the process for the whole range of parameters, except for boundary cases. 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 costaᵣenewal₂020. We then focus on the case p=3, extending the results derived for the p=2 case in a previous work CostaMaillardMuraro₂024. In particular, we show that the process may be stable even if one of the coefficients aᵢ is much greater than one.
Costa et al. (Tue,) studied this question.