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 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 limx↑1⁡ψ(x)=ψ1∈(0,∞), different limits of appropriately centered and scaled aggregated partial sums are shown to exist for β∈(−1,0), β=0, β∈(0,1) or β∈(1,∞), when taking first the limit as N→∞ and then the time scale n→∞, or vice versa. In fact, we give a partial solution to an open problem of Pilipauskaitė and Surgailis [13] by replacing the random-coefficient AR(1) process with a certain randomized INAR(1) process.