Markov Chains Indexed by Trees

Some limit results for Markov chains indexed by trees *

We consider a sequence of Markov chains (X)n=1,2,... with X = (X σ )σ∈T , indexed by the full binary tree T = T0 ∪ T1 ∪ . . . , where Tk is the k-th generation of T . In addition, let (Σk)k=0,1,2,... be a random walk on T with Σk ∈ Tk and R̃ = (R̃ t )t≥0 with R̃ t := XΣ[tn] , arising by observing the Markov chain X n along the random walk. We present a law of large numbers concerning the empirical...

un 2 00 3 Large deviations of Markov chains indexed by random trees

Given a finite typed rooted tree T with n vertices, the empirical subtree measure is the uniform measure on the n typed subtrees of T formed by taking all descendants of a single vertex. We prove a large deviation principle in n, with explicit rate function, for the empirical subtree measures of multitype Galton-Watson trees conditioned to have exactly n vertices. In the process, we extend the ...

Large deviations of Markov hains indexed by random trees

Some Limit Properties of the Harmonic Mean of Transition Probabilities for Markov Chains in Markovian Environments Indexed by Cayley’s Trees

A tree T is a graph which is connected and doesn’t contain any circuits. Given any two vertices α ̸ = β ∈ T, let αβ be the unique path connecting α and β. Define the graph distance d(α, β) to be the number of edges contained in the path αβ. Let T be an infinite tree with root 0.The set of all vertices with distance n from the root is called the nth generation of T, which is denoted by L n . We d...

Triplet Partially Markov Chains and Trees

Hidden Markov models (HMM), like chains or trees considered in this paper, are widely used in different situations. Such models, in which the hidden process X is a Markov one, allow one estimating the latter from an observed process Y , which can be seen as a noisy version of X . This is possible once the distribution of X conditional on Y is a Markov distribution. These models have been recent...