Scaling limit for supercritical nearly unstable Hawkes processes with heavy tail

In this paper, we establish the asymptotic behavior of { supercritical} nearly unstable Hawkes processes with a power law kernel. We find that, the Hawkes process in our context admits a similar equation to that in {MR3563196} for { subcritical} case. In particular, the rescaled Hawkes process (Zⁿₙₜ/n2α)t∈[0,1] converges in law to a kind of integrated fractional Cox Ingersoll Ross process with different coefficients from that in {MR3563196}, as n tends to infinity.