Concentration of Broadcast Models on Trees

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.