Some strong limit theorems for nonhomogeneous Markov chains indexed by controlled 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...

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

Some limit theorems for the second-order Markov chains indexed by a general infinite tree with uniform bounded degree

* 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 study a convergence theorem for a finite second-order Markov chain indexed by a general infinite tree with uniformly bounded degree. Meanwhile, the strong law of large numbers (LLN) and Shannon-...

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...

Ratio Limit Theorems for Markov Chains

Introduction. We consider Markov chains with stationary transition probabilities and state space S = 0, 1, 2, •• -, in both discrete and continuous time. As in [l ] the transition probabilities are denoted by pi}1 (indiscrete time) and pa(t) (in continuous time). We assume that the chains considered are irreducible and recurrent, and in addition that discrete time chains are aperiodic and that ...