Rearrangements of gaussian fields

TL;DR

This paper introduces multivariate rearrangement operators for Gaussian fields, proves their convergence, and explores how the limit depends on the geometric orientation of the domain, with applications in econometrics.

Contribution

It extends rearrangement techniques to multivariate Gaussian fields and establishes their almost sure convergence, considering geometric dependencies.

Findings

Rearrangement operators are generalized to multivariate Gaussian fields.

Almost sure convergence of these rearrangements is proven.

The limit depends on the orientation of simplices in the domain.

Abstract

The monotone rearrangement of a function is the non-decreasing function with the same distribution. The convex rearrangement of a smooth function is obtained by integrating the monotone rearrangement of its derivative. This operator can be applied to regularizations of a stochastic process to measure quantities of interest in econometrics. A multivariate generalization of these operators is proposed, and the almost sure convergence of rearrangements of regularized Gaussian fields is given. For the Fractional Brownian field or the Brownian sheet approximated on a simplicial grid, it appears that the limit object depends on the orientation of the simplices.