Markov chains on trees: almost lower and upper directed cases

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 $π$ 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 $π$ 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 Mötzkin 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.

Figures (4)

• Figure 1: $\varnothing$ is the root, and $u$ a node. Triangles figure finite or infinite subtrees, and disks, nodes. On the second picture, only the transitions from $u$ toward dark nodes are possible for almost upper-directed transitions matrices. On the third pictures, only the transitions from $u$ toward dark nodes are possible for ALD transitions matrix. For nearest-neighbor Markov chains, only, the neighbors of $u$ are accessible (fourth picture).
• Figure 2: Representation of the tree $t=\{\varnothing, 0,00,01,02,021,...\}$. The set of children of $02$ is $c_t(02)=\{021\}$, the nodes $1$, $00$, $01$ are leaves, $\varnothing, 0, 02, 021$ are internal nodes. On this example $\llbracket \varnothing,01\rrbracket=\{\varnothing, 0,01\}$.
• Figure 3: Each triangle represents a finite or infinite subtree. On the second picture, the blue edges (including those, not drawn, in the subtrees) are all oriented downwards: these edges are in all spanning trees rooted at $u$: the starting point of each oriented edge is any node which is not in $\llbracket \varnothing,u\rrbracket$. The set of blue edges form connected components, that are subtrees, each of them rooted a different vertices of $\llbracket \varnothing,u\rrbracket$ (this is the forest $F_u=(f_v,v\in\llbracket \varnothing,u\rrbracket)$). On the third picture, an example of what could be the set of edges going out from $\llbracket \varnothing,u\rrbracket$ of a spanning tree rooted at $u$: either directed to the parent, or to a node among their descendants. On the last picture, the "spanning tree condition" on $\llbracket \varnothing,u\rrbracket$. If we redirect each red edge toward the root of the blue component containing its second extremities, this forms a (red) spanning tree of $\llbracket \varnothing,u\rrbracket$: this condition is necessary and sufficient for the blue edges and red edges to form, together, a spanning tree of the global tree, rooted at $u$.
• Figure 4: To the right, the infinite tree $T$ which is constructed by grafting copies $(T_u)$ of the tree $\tau$ (to the left of the image) at each vertex of a copy of $\mathbb{N}$.

Theorems & Definitions (48)

• Definition 1.1
• Lemma 1.2
• proof
• Theorem 2.1
• Remark 2.2
• Remark 2.3: Inherent complexity.
• Proposition 2.4
• proof
• proof : Proof of \ref{['theo:1.3']}
• Lemma 2.5