Abstract
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 α ∈ (0, 1) and with idiosyncratic Poisson innovations. Assuming that α has a density function of the form ψ(x) (1 − x)
β
, x ∈ (0, 1), with β ∈ (−1, ∞) and
lim
x
↑
1
ψ
(
x
)
=
ψ
1
∈
(
0
,
∞
)
$\lim\limits_{x\uparrow 1} \psi(x) = \psi_1 \in (0, \infty)$
, different limits of appropriately centered and scaled aggregated partial sums are shown to exist for β ∈ (−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 β ∈ (0, ∞) 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.