Some remarks on scaling transition in limit theorems for random fields

TL;DR

This paper introduces simple linear random field examples on multi-dimensional integer lattices that demonstrate the scaling transition phenomenon, broadening understanding beyond previous finite-variance, long-range dependent cases.

Contribution

It provides new examples of linear random fields exhibiting scaling transition, including those with negative dependence and infinite variance, and offers a generalized definition of the phenomenon.

Findings

Scaling transition observed in fields with negative dependence

Examples include fields with infinite variance

Relation to Lamperti type theorems discussed

Abstract

In the paper we present simple examples of linear random fields defined on $\ZZ^2$ and $\ZZ^3$ which exhibit the scaling transition phenomenon. These examples lead to more general definition of the scaling transition and allow to understand the mechanism of appearance of this phenomenon better. In previous papers devoted to the scaling transition it was proved mainly for random fields with finite variance and long-range dependence. We consider random fields with finite and infinite variance. Our examples show that the scaling transition phenomenon can be observed for linear random fields with the so-called negative dependence, which is part of short-range dependence. Relation of the scaling transition with Lamperti type theorems for random fields is discussed.