# Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes with infinite variance

## TL;DR

This paper studies the combined effects of temporal and contemporaneous aggregation of random-coefficient AR(1) processes with infinite variance innovations, revealing diverse limit behaviors depending on tail and aggregation rates.

## Contribution

It extends previous work by analyzing the joint aggregation of processes with innovations in the domain of attraction of stable laws for all 0<α≤2, highlighting new limit behaviors.

## Key findings

- Diverse stable and non-stable limit behaviors depending on parameters.
- Extension of previous results from α=2 to all 0<α<2.
- Identification of conditions for different limit distributions.

## Abstract

We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of random-coefficient AR(1) process driven by i.i.d. innovations in the domain of normal attraction of an $\alpha$-stable distribution, $0< \alpha \le 2$, as both $N$ and the time scale $n$ tend to infinity, possibly at a different rate. Assuming that the tail distribution function of the random autoregressive coefficient regularly varies at the unit root with exponent $\beta > 0$, we show that, for $\beta < \max (\alpha, 1)$, the joint aggregate displays a variety of stable and non-stable limit behaviors with stability index depending on $\alpha$, $\beta$ and the mutual increase rate of $N$ and $n$. The paper extends the results of Pilipauskait\.e and Surgailis (2014) from $\alpha = 2$ to $0 < \alpha < 2$.