Iterated scaling limits for aggregation of randomized INAR(1) processes with idiosyncratic Poisson innovations

We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient $\alpha \in (0, 1)$ and with idiosyncratic Poisson innovations. Assuming that $\alpha$ has a density function of the form $\psi(x) (1 - x)^\beta$, $x \in (0, 1)$, with $\lim_{x\uparrow 1} \psi(x) = \psi_1 \in (0, \infty)$, different limits of appropriately centered and scaled aggregated partial sums are shown to exist for $\beta \in (-1, 0)$, $\beta = 0$, $\beta \in (0, 1)$ or $\beta \in (1, \infty)$, when taking first the limit as $N \to \infty$ and then the time scale $n \to \infty$, or vice versa. The paper extends some of the results in Pilipauskait\.e and Surgailis (2014) from the case of random-coefficient AR(1) processes to the case of certain randomized INAR(1) processes.