Markov chains on trees: almost lower and upper directed cases

TL;DR

This paper studies Markov chains on trees with specific directional constraints, establishing the existence of invariant measures, providing explicit criteria for recurrence, and introducing an efficient algorithm for measure computation.

Contribution

It introduces the concept of almost upper and lower-directed Markov chains on trees, proves the existence of invariant measures under irreducibility, and develops a leaf addition algorithm for their computation.

Findings

Invariant measures exist for almost upper-directed chains on infinite trees.

Explicit criteria for recurrence and positive recurrence are provided.

An efficient leaf addition algorithm enables measure computation without linear algebra.

Abstract

The transition matrix of a Markov chain $(X_k,k\geq 0)$ on a finite or infinite rooted tree is said to be almost upper-directed if, given $X_k$, the node $X_{k+1}$ is either a descendant of $X_k$ or the parent of $X_k$. It is said to be almost lower-directed if given $X_k$, $X_{k+1}$ is either an ancestor of $X_k$ or a child of $X_k$. These models include nearest neighbor Markov chains on trees. Under an irreducibility assumption, we show that every almost upper-directed transition matrix on infinite (locally finite) trees has some invariant measures. An invariant measure $\pi$ is expressed thanks to a determinantal formula. We give general explicit criteria for recurrence and positive recurrence. An efficient algorithm (the leaf addition algorithm) of independent interest allows $\pi$ to be computed on many trees, without resorting to linear algebra considerations. Flajolet, in a series of papers, provided some relations between continuous fractions, generating functions of weighted M\"otzkin paths, and used them in connection with the analysis of birth and death processes. These fruitful representations made it possible to establish many formulae for continuous fractions. Analogous considerations appear here: this type of study can be extended to weighted paths on trees, whose generating functions can also be expressed, this time in terms of multicontinuous fractions.