A Local Limit Theorem for Linear Random Fields

TL;DR

This paper proves a local limit theorem for linear random fields with independent innovations, covering cases with absolutely summable and square-summable coefficients, and includes new results for one-dimensional sequences and fractional processes.

Contribution

It introduces a local limit theorem for linear random fields under various coefficient conditions, extending existing results to higher dimensions and fractional processes.

Findings

Established local limit theorem for linear random fields.

Extended results to one-dimensional sequences.

Included simulation study for fractional processes.

Abstract

In this paper, we establish a local limit theorem for linear fields of random variables constructed from independent and identically distributed innovations each with finite second moment. When the coefficients are absolutely summable we do not restrict the region of summation. However, when the coefficients are only square-summable we add the variables on unions of rectangle and we impose regularity conditions on the coefficients depending on the number of rectangles considered. Our results are new also for the dimension 1, i.e. for linear sequences of random variables. The examples include the fractionally integrated processes for which the results of a simulation study is also included.