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February 2020 Operator-scaling Gaussian random fields via aggregation
Yi Shen, Yizao Wang
Bernoulli 26(1): 500-530 (February 2020). DOI: 10.3150/19-BEJ1133


We propose an aggregated random-field model, and investigate the scaling limits of the aggregated partial-sum random fields. In this model, each copy in the aggregation is a $\pm 1$-valued random field built from two correlated one-dimensional random walks, the law of each determined by a random persistence parameter. A flexible joint distribution of the two parameters is introduced, and given the parameters the two correlated random walks are conditionally independent. For the aggregated random field, when the persistence parameters are independent, the scaling limit is a fractional Brownian sheet. When the persistence parameters are tail-dependent, characterized in the framework of multivariate regular variation, 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.


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Yi Shen. Yizao Wang. "Operator-scaling Gaussian random fields via aggregation." Bernoulli 26 (1) 500 - 530, February 2020.


Received: 1 February 2018; Revised: 1 May 2019; Published: February 2020
First available in Project Euclid: 26 November 2019

zbMATH: 07140507
MathSciNet: MR4036042
Digital Object Identifier: 10.3150/19-BEJ1133

Keywords: Aggregation , Fractional Brownian sheet , functional central limit theorem , Gaussian random field , long-range dependence , operator-scaling property

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability


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Vol.26 • No. 1 • February 2020
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