  We investigate the asymptotic behavior as time goes to infinity of Hawkes
processes whose regression kernel has $L^1$ norm close to one and power law
tail of the form $x^{-(1+\alpha)}$, with $\alpha\in(0,1)$. We in particular
prove that when $\alpha\in(1/2,1)$, after suitable rescaling, their law
converges to that of a kind of integrated fractional Cox-Ingersoll-Ross
process, with associated Hurst parameter $H=\alpha-1/2$. This result is in
contrast to the case of a regression kernel with light tail, where a classical
Brownian CIR process is obtained at the limit. Interestingly, it shows that
persistence properties in the point process can lead to an irregular behavior
of the limiting process. This theoretical result enables us to give an
agent-based foundation to some recent findings about the rough nature of
volatility in financial markets.
