[Paper Review] Operator-scaling Gaussian random fields via aggregation

Yi Shen, Yizao Wang|arXiv (Cornell University)|Dec 19, 2017

Stochastic processes and statistical mechanics29 references3 citations

TL;DR

This paper proposes an aggregated random-field model using ±1-valued fields built from two correlated one-dimensional random walks with random persistence parameters. By analyzing scaling limits under different dependence structures of the parameters, the study establishes that operator-scaling Gaussian random fields, including fractional Brownian sheets and a broad class of long-range dependent fields, emerge as limits depending on the growth rates of the rectangular domain, with critical speed yielding a rich family of long-memory processes.

ABSTRACT

We propose an aggregated random-field model, and investigate the scaling limits of the aggregated partial-sum random fields. In our model, each copy of the random field in the aggregation is built from two correlated one-dimensional random walks, each with a random persistence parameter. When the persistence parameters are independent, the scaling limit is a fractional Brownian sheet. When the persistence parameters are dependent, the scaling limit is more delicate, and in particular depends on the growth rates of the underlying rectangular region along two directions: at different rates different operator-scaling Gaussian random fields appear as the region area tends to infinity. In particular, at the so-called critical speed, a large family of Gaussian random fields with long-range dependence arise in the limit. We also identify four different regimes at non-critical speed where fractional Brownian sheets arise in the limit.

Motivation & Objective

• To develop a spatial aggregation model that generates operator-scaling Gaussian random fields as scaling limits.
• To investigate how dependence structures in persistence parameters affect the limiting behavior of aggregated random fields.
• To identify conditions under which long-range dependence arises in the limit, particularly at critical growth rates of the domain.
• To extend the functional central limit theorem framework to two-dimensional spatial models with anisotropic scaling.
• To characterize the emergence of fractional Brownian sheets and more general operator-scaling fields under different parameter dependence regimes.

Proposed method

• Aggregates i.i.d. copies of ±1-valued random fields constructed from two correlated one-dimensional random walks with random persistence parameters.
• Introduces a flexible joint distribution for the persistence parameters, allowing for independence or tail dependence via multivariate regular variation.
• Applies functional central limit theorem techniques to study the scaling limits of partial-sum fields over rectangular domains.
• Uses spectral analysis and integral transforms to derive the asymptotic covariance structure of the limiting process.
• Derives conditions under which the limit is a fractional Brownian sheet (when parameters are independent) or more general operator-scaling Gaussian fields (when parameters are tail-dependent).
• Employs beta function identities and change-of-variables to evaluate divergent integrals in the limit derivation, particularly for the critical speed regime.

Experimental results

Research questions

• RQ1 Under what conditions does the aggregation of dependent random walks lead to operator-scaling Gaussian random fields in the limit?
• RQ2 How does the dependence structure of the persistence parameters—specifically tail dependence—affect the scaling limit?
• RQ3 What happens to the limiting process when the rectangular domain grows at different rates along the two spatial dimensions?
• RQ4 Can the critical growth rate of the domain lead to a broad family of long-range dependent Gaussian fields?
• RQ5 What is the precise form of the asymptotic covariance structure in the limit, and how does it relate to the parameters of the underlying random walks?

Key findings

• When the persistence parameters are independent, the scaling limit of the aggregated random field is a fractional Brownian sheet.
• When the persistence parameters are tail-dependent and grow at non-critical rates, four distinct regimes emerge, each yielding a fractional Brownian sheet in the limit.
• At the critical growth rate, a large family of operator-scaling Gaussian random fields with long-range dependence arises as the limit.
• The limit process depends on the joint distribution of the persistence parameters and the anisotropic growth rates of the rectangular domain.
• The asymptotic covariance structure of the limit is derived using spectral analysis and integral transforms, with explicit expressions involving the beta function and parameters α₁, α₂, and H.
• The convergence of the finite-dimensional distributions and the tightness of the finite-dimensional distributions are established via dominated convergence and moment bounds in the supplementary material.