Aggregation of autoregressive processes and long memory

TL;DR

This paper investigates how aggregating autoregressive and Ornstein-Uhlenbeck processes can lead to long memory effects, emphasizing the role of pole distribution and spectral density mixtures in this phenomenon.

Contribution

It extends existing theories by analyzing the impact of pole concentration and angular dispersion on long memory creation in aggregated processes.

Findings

Long memory depends on poles' proximity to stability boundary.

Angular dispersion of poles influences long memory emergence.

Aggregation of complex processes can induce long-range dependence.

Abstract

We study the aggregation of AR processes and generalized Ornstein-Uhlenbeck (OU) processes. Mixture of spectral densities with random poles are the main tool. In this context, we apply our results for the aggregation of doubly stochastic interactives processes, see Dacunha-Castelle and Fermin (2006). Thus, we study the relationship between aggregation of autoregressive processes and long memory considering complex interaction structures. We precise a very interesting qualitative phenomena: how the long memory creation depends on the poles concentration near to the boundary of stability (measured in the Prokhorov sense). Our results extends the results given by Oppenheim and Viano (2004), and highlight the importance of the angular dispersion measure of poles in the appearance of the long memory.