Abstract
A class of small-deviation theorems for the relative entropy densities of arbitrary random field on the generalized Bethe tree are discussed by comparing the arbitrary measure with the Markov measure on the generalized Bethe tree. As corollaries, some Shannon-Mcmillan theorems for the arbitrary random field on the generalized Bethe tree, Markov chain field on the generalized Bethe tree are obtained.
Highlights
Introduction and LemmaLet T be a tree which is infinite, connected and contains no circuits
Ye and Berger see 6 have studied the ergodic property and ShannonMcMillan theorem for PPG-invariant random fields on trees
Yang et al 7–9 have recently studied a.s. convergence of Shannon-McMillan theorems, the limit properties and the asymptotic equipartition property for Markov chains indexed by a homogeneous tree and the Cayley tree, respectively
Summary
Let T be a tree which is infinite, connected and contains no circuits. Given any two vertices x / y ∈ T, there exists a unique path x x1, x2, . . . , xm y from x to y with x1, x2, . . . , xm distinct. With the development of the information theory scholars get to study the Shannon-McMillan theorems for the random field on the tree graph see 4. Ye and Berger see 6 have studied the ergodic property and ShannonMcMillan theorem for PPG-invariant random fields on trees. Their results only relate to the convergence in probability. Yang et al 7–9 have recently studied a.s. convergence of Shannon-McMillan theorems, the limit properties and the asymptotic equipartition property for Markov chains indexed by a homogeneous tree and the Cayley tree, respectively. A class of Shannon-McMillan theorems for arbitrary random fields and the Markov chains field on the generalized Bethe tree are obtained. Φ μ | μQ can be look on as a type of measures of the deviation between the arbitrary random fields and the Markov chain fields on the generalized Bethe tree
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