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 $\beta\in(1,\infty)$, when taking first the limit as $N\to\infty$ and then the time scale $n\to\infty$, 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.