Limit theorems for functionals of linear processes in critical regions

Let X={X_n:n∈N} be the linear process defined by X_n=∑_j=1^∞a_jɛ_n−j, where the coefficients a_j=j^−βℓ(j) are constants with β>0 and ℓ a slowly varying function, and the innovations {ɛ_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]=∑_n=1^[Nt][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.