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

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 $\beta \in (-1,\mathrm{\infty })$ and ${\lim }_{x\uparrow 1}\psi (x)={\psi }_1\in (0,\mathrm{\infty })$, 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 $\beta \in (0,\mathrm{\infty })$ 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.