Limit theorems for functionals of linear processes in critical regions

2025/02/28 by Xiong, Yudan, Xu, Fangjun, Yu, Jinjiong
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2502.20956

Abstract

Let X=\X_n: n∈ℕ\ be the linear process defined by X_n=∑^∞_j=1 a_jε_n-j, where the coefficients a_j=j^-βℓ(j) are constants with β>0 and ℓ a slowly varying function, and the innovations \ε_n\_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]=∑^[Nt]_n=1[K(X_n)-𝔼K(X_n)] 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.

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