Iterated scaling limits for aggregation of random coefficient AR(1) and INAR(1) processes

TL;DR

This paper investigates the limiting behavior of aggregated random coefficient AR(1) and INAR(1) processes, revealing different Brownian limits depending on the order of limits taken, and completes previous work by analyzing the critical case where eta=1.

Contribution

It provides a comprehensive analysis of iterated scaling limits for aggregated processes, specifically addressing the critical case eta=1, which was previously unresolved.

Findings

Different Brownian limit processes are identified for eta=1.

The order of limits (N rrow N and n rrow n) affects the limiting process.

Completes the analysis of scaling limits for all parameter values in the models.

Abstract

We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of strictly stationary AR(1) and INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient $\alpha \in (0, 1)$ and idiosyncratic 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 Brownian limit processes of appropriately centered and scaled aggregated partial sums are shown to exist in case $\beta=1$ when taking first the limit as $N \to \infty$ and then the time scale $n \to \infty$, or vice versa. This paper completes the one of Pilipauskait\.e and Surgailis (2014), and Barczy, Ned\'enyi and Pap (2015), where the iterated limits are given for every other possible value of the parameter $\beta$ for the two types of models.