Concentration of Broadcast Models on Trees

TL;DR

This paper extends Marton's concentration inequality to broadcast models on finite trees, providing conditions under which marginals form a normal Levy family based on Lipschitz constants and tree growth.

Contribution

It introduces a new concentration inequality for broadcast models on trees and characterizes when marginals form a Levy family.

Findings

Established a concentration inequality for broadcast models on finite trees.

Derived conditions for marginals to form a normal Levy family.

Linked Lipschitz constants and tree growth to concentration properties.

Abstract

An inequality of K. Marton shows that the joint distribution of a Markov chain with uniformly contracting transition kernels exhibits concentration. We prove an analogous inequality for broadcast models on finite trees. We use this inequality to develop a condition for the sequence of depth-$k$ marginals of a broadcast model on a rooted infinite tree to form a normal L\'{e}vy family in terms of the Lipschitz constants of the transition kernels and the growth rate of the tree.