Scaling limit of heavy tailed nearly unstable INAR(∞) processes and rough fractional diffusions

In this paper, we investigated the scaling limit of heavy tailed unstable INAR(∞) processes where the ¹ norm of the kernel vector is close to $1$ and have power law tail of the form n-(1+α), with α∈(12,1). The result is in contrast to the continuous-time heavy tailed unstable Hawkes processes and light-tailed INAR(p) processes. We show that the discrete-time scaling limit also has long-memory property and can also be seen as an integrated fractional Cox-Ingersoll-Ross process.