Random Fractals and Tree - Indexed Markov

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...

Some Limit Properties of Random Transition Probability for Second-Order Nonhomogeneous Markov Chains Indexed by a Tree

A class of small deviation theorems for the random fields on an m rooted Cayley tree

* Correspondence: [email protected] Faculty of Science, Jiangsu University, Zhenjiang 212013, China Full list of author information is available at the end of the article Abstract In this paper, we are to establish a class of strong deviation theorems for the random fields relative to mth-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree. As corollaries, we obtain the st...

Tree-Structured Like Representations for Continuous and Discrete Indexed Markov Random Fields

Markov random fields (MRFs) are ubiquitous models for conveniently capturing complex statistical dependencies given a collection of random variables. For example, when modeling the temperature distribution of a continuously indexed region, it is common to assume that the temperature at one point depends on some differential neighborhood. MRFs can capture such dependencies using a set of conditi...

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...