Scaling limit of heavy tailed nearly unstable cumulative INAR($\infty$) processes and rough fractional diffusions

Abstract

In this paper, we investigate the scaling limit of heavy-tailed nearly unstable cumulative INAR($\infty$) processes. These processes exhibit a power-law tail of the form $n^{-(1+\hat{I}\pm )}$ for $\hat{I}\pm \in (\frac{1}{2},1)$, and the ${\ell }^1$ norm of the kernel vector converges to 1. We demonstrate that the discrete-time scaling limit retains a long-memory property and can be viewed as an integrated fractional Cox-Ingersoll-Ross process. Moreover, we present an efficient method for simulating the fractional Cox-Ingersoll-Ross process. The simulation and Goodness-of-Fit Test code are available at https://github.com/gagawjbytw/INAR-rough-Heston.