  In this paper, we investigated the scaling limit of heavy-tailed unstable
cumulative INAR($\infty$) processes. These processes exhibit a power law tail
of the form $n^{-(1+\alpha)}$, with $\alpha \in (\frac{1}{2}, 1)$, where the
$\ell^1$ norm of the kernel vector is close to $1$. The result is in contrast
to scaling limit of the continuous-time heavy tailed unstable Hawkes processes
and the one of 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.
