Gaussian vectors with Markov property

TL;DR

This paper explores the properties of Gaussian vectors with the Markov property, establishing their geometric and probabilistic characteristics, and providing a Euclidean proof of entropy maximality for such vectors.

Contribution

It offers new Euclidean geometric characterizations of Markov Gaussian vectors and a simplified proof of their differential entropy maximality.

Findings

Equivalent Euclidean and probabilistic characterizations of Markov Gaussian vectors

Euclidean proof of differential entropy maximality for Markov Gaussian vectors

Enhanced understanding of Gaussian vectors' geometric properties

Abstract

We demonstrate the parallel between the properties of Gaussian vectors and the Euclidean geometry. In particular we study the Markov property and give various equivalent Euclidean and probabilistic characterizations. We also give a simple Euclidean proof of the conditional maximality of the differential entropy for the Markov Gaussian vector (related to the Burg's Theorem).