From random partitions to fractional Brownian sheets

TL;DR

This paper introduces discrete random-field models based on random partitions of the plane, demonstrating that fractional Brownian sheets emerge as limits, thus providing a new discrete analogue for these continuous stochastic processes.

Contribution

The paper develops a novel class of discrete random-field models linked to random partitions, establishing their convergence to fractional Brownian sheets with a full range of Hurst indices.

Findings

Fractional Brownian sheets arise as limits of the proposed models.

The models serve as discrete analogues of fractional Brownian sheets.

Functional central limit theorems are established for the models.

Abstract

We propose discrete random-field models that are based on random partitions of $\mathbb{N}^2$. The covariance structure of each random field is determined by the underlying random partition. Functional central limit theorems are established for the proposed models, and fractional Brownian sheets, with full range of Hurst indices, arise in the limit. Our models could be viewed as discrete analogues of fractional Brownian sheets, in the same spirit that the simple random walk is the discrete analogue of the Brownian motion.