Dynkin's Isomorphism with Sign Structure

TL;DR

This paper extends Dynkin's isomorphism to include sign structures, enabling Gaussian fields with arbitrary covariance sign patterns, which broadens its applicability in prediction and time series analysis.

Contribution

The authors introduce a sign-structured extension of Dynkin's isomorphism, allowing for Gaussian fields with general covariance sign patterns, unlike the original non-negative covariance restriction.

Findings

Enables Gaussian fields with arbitrary covariance signs

Broadens applications in prediction and time series analysis

Maintains the core properties of Dynkin's isomorphism

Abstract

The Dynkin isomorphism associates a Gaussian field to a Markov chain. These Gaussian fields can be used as priors for prediction and time series analysis. Dynkin's construction gives Gaussian fields with all non-negative covariances. We extend Dynkin's construction (by introducing a sign structure on the Markov chain) to allow general covariance sign patterns.