The Asymptotic Equipartition Property for Nonhomogeneous Markov Chains Indexed by a Homogeneous Tree

A class of small deviation theorems for functionals of random fields on a homogeneous tree

“…Assume that μ(x T (n) ) is always strictly positive. Let ϕ n (ω) = μ(X T (n) ) P (X T (n) ) , (6) ϕ(ω) = lim sup n→∞ 1 |T (n) | ln ϕ n (ω), (7) ϕ(ω) will be called the asymptotic logarithmic likelihood ratio.…”

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

“…Then {t n (λ, ω), F n , n ≥ 1} is a nonnegative martingale. Proof: The proof of Lemma 1 is similar to Lemma 2.1 in [9], so we omit it. Theorem 1 Let {X t , t T} and {g t (x, y, z), t T} be defined as Lemma 1,…”

“…Takacs (see [8]) has studied the strong law of large numbers for the univariate functions of finite Markov chains indexed by an infinite tree with uniformly bounded degree. Recently, Yang (see [9]) has studied the strong law of large numbers and Shannon-McMillan theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Huang and Yang (see [10]) have studied the strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree, which generalize the result of [8].…”

Markov approximation of arbitrary random field on homogeneous trees

“…Recently, Yang [5] have studied some strong limit theorems for countable homogeneous Markov chains indexed by a homogeneous tree and the strong law of large numbers and the AEP for finite homogeneous Markov chains indexed by a homogeneous tree. Yang and Ye [6] have studied strong theorems for countable nonhomogeneous Markov chains indexed by a homogeneous tree and the strong law of large numbers and the AEP for finite nonhomogeneous Markov chains indexed by a homogeneous tree. Small deviation theorems are a class of strong limit theorems expressed by inequalities.…”