We prove a law of large numbers and a functional central limit theorem for
multivariate Hawkes processes observed over a time interval $[0,T]$ in the
limit $T \rightarrow \infty$. We further exhibit the asymptotic behaviour of
the covariation of the increments of the components of a multivariate Hawkes
process, when the observations are imposed by a discrete scheme with mesh
$\Delta$ over $[0,T]$ up to some further time shift $\tau$.

We consider the problem of adaptive estimation of the regression function in
a framework where we replace ergodicity assumptions (such as independence or
mixing) by another structural assumption on the model. Namely, we propose
adaptive upper bounds for kernel estimators with data-driven bandwidth
(Lepski's selection rule) in a regression model where the noise is an increment
of martingale. It includes, as very particular cases, the usual i.i.d.
regression and auto-regressive models.