On the asymptotic behavior of the Diaconis and Freedman's chain in a multidimensional simplex

TL;DR

This paper studies the long-term behavior of Diaconis and Freedman's chain within a multidimensional simplex, establishing conditions for invariant measures, explicit densities in special cases, and classifying behaviors in two dimensions.

Contribution

It introduces a new setting for the chain in higher dimensions, provides conditions for invariant measures, and classifies its behavior in two dimensions.

Findings

Existence and uniqueness of invariant measures under certain conditions

Explicit formulas for invariant densities in specific cases

Complete classification of the chain's behavior in two dimensions

Abstract

In this paper, we give out a setting of an Diaconis and Freedman's chain in a multidimensional simplex and consider its asymptotic behavior. By using techniques in random iterated functions theory and quasi-compact operators theory, we first give out some sufficient conditions which ensure the existence and uniqueness of an invariant probability measure. In some particular cases, we give out explicit formulas of the invariant probability density. Moreover, we completely classify all behaviors of this chain in dimensional two. Eventually, some other settings of the chain are discussed.