Ergodic theorem for branching Markov chains indexed by trees with arbitrary shape

Julien Weibel

Abstract:

We prove an ergodic theorem for Markov chains indexed by the Ulam-Harris-Neveu tree
over large subsets with arbitrary shape under two assumptions: with high probability, two
vertices in the large subset are far from each other and have their common ancestor close
to the root. The assumption on the common ancestor can be replaced by some regularity
assumption on the Markov transition kernel. We verify that those assumptions are satisfied
for some usual trees. Finally, with Markov-Chain Monte-Carlo considerations in mind, we
prove when the underlying Markov chain is stationary and reversible that the Markov chain,
that is the line graph, yields minimal variance for the empirical average estimator among
trees with a given number of nodes.