Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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.

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

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.
Paper Structure (8 sections, 11 theorems, 34 equations, 4 figures)

This paper contains 8 sections, 11 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Notations
  4. Statement of the main result
  5. Examples satisfying Assumptions \ref{['assump:An_ergodic_theorem']} and \ref{['assump:ancestor_tight']}
  6. Some simple deterministic trees
  7. Super-critical Bienaymé-Galton-Watson trees
  8. Dependence of the variance on the shape of the tree

Key Result

Theorem 1.2

Let $(A_n)_{n\in\mathbb{N}}$ be a sequence of finite subsets of $\mathbb{T}^\infty$ that satisfies Assumption assump:An_ergodic_theorem. Let $X$ be a branching Markov process indexed by $\mathbb{T}^\infty$ with values in $\mathcal{X}$ whose transition kernel $Q$ is ergodic. Assume that either $Q$ is

Figures (4)

  • Figure 1: Comparison of the double-cherry graph and the line graph ($n=6$); both graphs have an exactly balanced bipartite $2$-coloring, and thus satisfy $H_T(-1)=0$.
  • Figure 2: The unrooted trees $T$ and $T'$ for $\alpha\in(0,1)$
  • Figure 3: The unrooted tree $T$ before modification
  • Figure 4: The unrooted tree $T'$ in Case 3 after modification

Theorems & Definitions (24)

  • Remark 1.1: Some sufficient conditions for Assumptions \ref{['assump:An_ergodic_theorem']} and \ref{['assump:ancestor_tight']}, see Section \ref{['section:sufficient_conditions']}
  • Theorem 1.2: Ergodic theorem for Markov processes on trees with arbitrary shape
  • Remark 1.3
  • Proposition 1.4: The line graph has minimal variance
  • Lemma 1.5: The line graph minimizes the Hosoya-Wiener polynomial
  • Definition 2.1: Markov process
  • Theorem 2.2: Ergodic theorem for Markov processes on trees with arbitrary shape
  • proof
  • Lemma 2.3
  • proof
  • ...and 14 more