On simultaneous limits for aggregation of stationary randomized INAR(1) processes with Poisson innovations. (English) Zbl 1478.60085
Summary: 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 \(\lim\limits_{x\uparrow 1}\psi(x)=\psi_1\in (0, \infty)\), different limits of appropriately centered and scaled aggregated partial sums are shown to exist for \(\beta\in (-1, 0]\) 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.
MSC:
| 60F05 | Central limit and other weak theorems |
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
| 60E10 | Characteristic functions; other transforms |
| 60G52 | Stable stochastic processes |
| 60G15 | Gaussian processes |
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