Limit theorems for linear random fields with innovations in the domain of attraction of a stable law

TL;DR

This paper establishes limit theorems for linear random fields with innovations in the domain of attraction of stable laws, covering cases with infinite variance and various summability conditions on the coefficients.

Contribution

It extends the theory of limit theorems to linear random fields with stable-like innovations, including both absolutely summable and non-summable coefficient cases.

Findings

Proves convergence in distribution for partial sums of such fields.

Establishes local limit theorems under broad conditions.

Handles innovations with infinite second moments in the domain of attraction.

Abstract

In this paper we study the convergence in distribution and the local limit theorem for the partial sums of linear random fields with i.i.d. innovations that have infinite second moment and belong to the domain of attraction of a stable law with index $0<\alpha\leq2$ under the condition that the innovations are centered if $1<\alpha\leq2$ and are symmetric if $\alpha=1$. We establish these two types of limit theorems as long as the linear random fields are well-defined, the coefficients are either absolutely summable or not absolutely summable.