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April 2017 Invariance principles for operator-scaling Gaussian random fields
Hermine Biermé, Olivier Durieu, Yizao Wang
Ann. Appl. Probab. 27(2): 1190-1234 (April 2017). DOI: 10.1214/16-AAP1229

Abstract

Recently, Hammond and Sheffield [Probab. Theory Related Fields 157 (2013) 691–719] introduced a model of correlated one-dimensional 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 $d\geq2$. 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.

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Hermine Biermé. Olivier Durieu. Yizao Wang. "Invariance principles for operator-scaling Gaussian random fields." Ann. Appl. Probab. 27 (2) 1190 - 1234, April 2017. https://doi.org/10.1214/16-AAP1229

Information

Received: 1 January 2016; Revised: 1 June 2016; Published: April 2017
First available in Project Euclid: 26 May 2017

zbMATH: 1370.60057
MathSciNet: MR3655864
Digital Object Identifier: 10.1214/16-AAP1229

Subjects:
Primary: 60F17
Secondary: 60G18, 60G22, 60G60

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.27 • No. 2 • April 2017
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