Invariance principles for operator-scaling Gaussian random fields

TL;DR

This paper studies the scaling limits of a generalized class of operator-scaling Gaussian random fields derived from correlated random walks on multi-dimensional integer lattices, revealing different limit behaviors depending on the growth of rectangular sets.

Contribution

It introduces a new model of $ ext{Z}^d$-indexed random fields with dependence governed by an underlying graph, analyzing their scaling limits and the emergence of operator-scaling Gaussian fields.

Findings

Different limit fields arise depending on the growth rates of sets in various directions.

In a critical regime, the limit field fully reflects the dependence structure of the discrete model.

Limit fields exhibit diverse path and increment properties, including invariance and independence in some directions.

Abstract

Recently, Hammond and Sheffield introduced a model of correlated random walks that scale to fractional Brownian motions with long-range dependence. In this paper, we consider a natural generalization of this model to dimension $d\geq 2$. We define a $\mathbb Z^d$-indexed random field with dependence relations governed by an underlying random graph with vertices $\mathbb Z^d$, and we study the scaling limits of the partial sums of the random field over rectangular sets. An interesting phenomenon appears: depending on how fast the rectangular sets increase along different directions, different random fields arise in the limit. In particular, there is a critical regime where the limit random field is operator-scaling and inherits the full dependence structure of the discrete model, whereas in other regimes the limit random fields have at least one direction that has either invariant or independent increments, no longer reflecting the dependence structure in the discrete model. The limit random fields form a general class of operator-scaling Gaussian random fields. Their increments and path properties are investigated.