Limit theorems for functionals of linear processes in critical regions

Yudan Xiong, Fangjun Xu and Jinjiong Yu

Stochastic Processes and their Applications, 2026, vol. 191, issue C

Abstract: Let X={Xn:n∈N} be the linear process defined by Xn=∑j=1∞ajɛn−j, where the coefficients aj=j−βℓ(j) are constants with β>0 and ℓ a slowly varying function, and the innovations {ɛn}n∈Z are i.i.d. random variables belonging to the domain of attraction of an α-stable law with α∈(0,2]. Limit theorems for the partial sum S[Nt]=∑n=1[Nt][K(Xn)−EK(Xn)] with proper measurable functions K have been extensively studied, except for two critical regions: I. α∈(1,2),β=1 and II. αβ=2,β≥1. In this paper, we address these open scenarios and identify the asymptotic distributions of S[Nt] under mild conditions.

Keywords: Linear process; Long/short memory; Limit theorem; Domain of attraction of stable law (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1016/j.spa.2025.104784

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