Scaling limits for INAR$(\infty)$ processes

TL;DR

This work analyzes INAR$(\infty)$ processes, unifying discrete-time Hawkes processes and INAR$(1)$ under a single infinite-order thinning framework. It derives a large deviations principle, a moderate deviations principle, a law of large numbers, and a central limit theorem for the scaled partial sums $\tfrac{1}{n}\sum_{k=1}^n X_k$, with explicit rate functions and variance formulas; the results hinge on cumulant generating functions and a Gärtner–Ellis type approach. The LDP uses the concave function $F(x)=x-\sum_{k=1}^{\infty}\log\mathbb{E}[e^{x\xi_k}]$ and its smaller root $f_{\infty}(\theta)$, yielding $I(x)=\sup_{\theta\le\theta_c}\{\theta x-\log\mathbb{E}[e^{f_{\infty}(\theta)\varepsilon}]\}$, while the MDP yields a quadratic rate function $J(x)$. Special cases recover known results for Hawkes and INAR$(1)$ models, and the analysis extends to nearly unstable regimes through explicit moment conditions. Overall, the paper fills a gap by establishing deviation principles for INAR$(\infty)$ processes and linking discrete-time counting processes to Hawkes theory.

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}.

Theorems & Definitions (40)

• Theorem 2.1
• Remark 2.2
• Corollary 2.3
• Remark 2.4
• Corollary 2.5
• Remark 2.6
• Theorem 2.7
• Remark 2.8
• Corollary 2.9
• Remark 2.10