On the central limit theorem for stationary random fields under L 1 -projective condition

TL;DR

This paper investigates the extension of the central limit theorem for ergodic stationary random fields under L1-projective conditions, revealing that additional assumptions are necessary for the theorem to hold.

Contribution

It identifies the limitations of existing CLT generalizations under L1-projective criteria and introduces the need for extra conditions in the context of stationary random fields.

Findings

The CLT cannot be fully extended under L1-projective conditions without additional assumptions.

Additional conditions are necessary for the CLT to hold for ergodic stationary random fields.

The study clarifies the boundaries of current CLT generalizations in this setting.

Abstract

The first aim of this paper is to wonder to what extent we can generalize the central limit theorem of Gordin [5] under the so-called L 1-projective criteria to ergodic stationary random fields when completely commuting filtrations are considered. Surprisingly it appears that this result cannot be extended to its full generality and that an additional condition is needed.