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We investigate joint temporal and contemporaneous aggregation of Nindependent copies of strictly stationary INteger-valued AutoRegressiveprocesses of order 1 (INAR(1)) with random coefficient $\alpha\in(0,1)$ andwith idiosyncratic Poisson innovations. Assuming that $\alpha$ has a densityfunction of the form $\psi(x) (1 - x)^\beta$, $x \in (0,1)$, with$\beta\in(-1,\infty)$ and $\lim_{x\uparrow 1} \psi(x) = \psi_1 \in (0,\infty)$,different limits of appropriately centered and scaled aggregated partial sumsare shown to exist for $\beta\in(-1,0]$ in the so-called simultaneous case,i.e., when both $N$ and the time scale $n$ increase to infinity at a givenrate. The case $\beta\in(0,\infty)$ remains open. We also give a new explicitformula for the joint characteristic functions of finite dimensionaldistributions of the appropriately centered aggregated process in question.

On simultaneous limits for aggregation of stationary randomized INAR(1) processes with Poisson innovations