Scaling limits for INAR$(\infty)$ processes

TL;DR

This paper investigates the asymptotic behavior of INAR(∞) processes, including LLN, CLT, and deviation principles, unifying and extending results for related processes like Hawkes and INAR(1).

Contribution

It provides a comprehensive analysis of INAR(∞) processes, deriving new limit theorems and deviations that encompass and extend previous results for simpler models.

Findings

Established LLN, CLT, large and moderate deviations for INAR(∞) processes.

Unified framework covering discrete-time Hawkes and INAR(1) processes.

Recovered and extended existing deviation results for related models.

Abstract

In this paper, we study law of large numbers, central limit theorem, large and moderate deviations for INAR($\infty$) processes, which as a special case, includes both discrete-time linear Hawkes process and INAR(1) process in the literature. Our results recover existing results on large and moderate deviations for the discrete-time Hawkes process as studied in \cite{Wang2} and for the INAR(1) process as in \cite{Yu}.