Iterated limits for aggregation of randomized INAR(1) processes with Poisson innovations

TL;DR

This paper investigates the asymptotic behavior of aggregated sums of independent randomized INAR(1) processes with Poisson innovations, revealing different limit regimes based on the distribution of the random coefficient.

Contribution

It extends previous work by analyzing the joint limits of aggregated INAR(1) processes with random coefficients, providing partial solutions to an open problem in the field.

Findings

Different limit behaviors depending on the parameter ta

Existence of various limit theorems for aggregated sums

Partial resolution of an open problem by replacing AR(1) with INAR(1)

Abstract

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 $\alpha\in(0,1)$ 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 $\lim_{x\uparrow 1}\psi(x) = \psi_1 \in(0,\infty)$, different limits of appropriately centered and scaled aggregated partial sums are shown to exist for $\beta\in(-1,0)$, $\beta = 0$, $\beta\in(0,1)$ or $\beta\in(1,\infty)$, when taking first the limit as $N\to\infty$ and then the time scale $n\to\infty$, or vice versa. In fact, we give a partial solution to an open problem of Pilipauskaite and Surgailis (2014) by replacing the random-coefficient AR(1) process with a certain randomized INAR(1) process.