  We investigate 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
$\beta\in(-1,\infty)$ and $\lim_{x\uparrow 1} \psi(x) = \psi_1 \in (0,\infty)$,
different limits of appropriately centered and scaled aggregated partial sums
are 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 given
rate. The case $\beta\in(0,\infty)$ remains open. We also give a new explicit
formula for the joint characteristic functions of finite dimensional
distributions of the appropriately centered aggregated process in question.
