Statistical inference versus mean field limit for Hawkes processes

We consider a population of N individuals, of which we observe the number actions as time evolves. For each couple of individuals $(i,j)$, j may or influence i, which we model by i.i.d. Bernoulli$(p)$-random variables, some unknown parameter p\∈ (0,1]. Each individual acts autonomously at unknown rate \μ>0 and acts by mimetism at some rate depending on the of recent actions of the individuals which influence him, the age of actions being taken into account through an unknown function \φ(roughly, decreasing and with fast decay). The goal of this paper is to p, which is the main charateristic of the graph of interactions, in asymptotic N\→\∞, t\→\∞. The main issue is that the mean limit (as N \→ \∞) of this model is unidentifiable, in that it depends on the parameters \μ and p\φ. Fortunately, this mean limit is not valid for large times. We distinguish the subcritical case,, roughly, the mean number mₜ of actions per individual increases and the supercritical case, where mₜ increases exponentially. the nuisance parameter \φ is non-parametric, we are able, in cases, to estimate p without estimating \φ in a nonparametric, with a precision of order N-1/2+N1/2mₜ⁻¹, up to some small loss. We explain, using a Gaussian toy model, the reason why rate of convergence might be (almost) optimal.