  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 $\tau$. The behaviour of
this functional depends on the relative size of $\Delta$ and $\tau$ with
respect to $T$ and enables to give a full account of the second-order
structure. As an application, we develop our results in the context of
financial statistics. We introduced in a previous work a microscopic stochastic
model for the variations of a multivariate financial asset, based on Hawkes
processes and that is confined to live on a tick grid. We derive and
characterise the exact macroscopic diffusion limit of this model and show in
particular its ability to reproduce important empirical stylised fact such as
the Epps effect and the lead-lag effect. Moreover, our approach enable to track
these effects across scales in rigorous mathematical terms.
