Iterated limits for aggregation of randomized INAR(1) processes with 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 $\beta \in (1,\mathrm{\infty })$, when taking first the limit as $N\to \mathrm{\infty }$ and then the time scale $n\to \mathrm{\infty }$, or vice versa. In fact, we give a partial solution to an open problem of Pilipauskaite and Surgailis (2014) by replacing the random-coefficient AR(1) process with a certain randomized INAR(1) process.