A class of strong deviation theorems for the random fields associated with bifurcating Markov chains indexed by a binary tree

Abstract

In this paper, we first introduce the asymptotic logarithmic likelihood ratio as a measure of the deviation between the arbitrary random fields and the bifurcating Markov chain on a binary tree. Then a class of strong deviation theorems for the random fields associated with bifurcating Markov chains indexed by a binary tree is established by constructing a nonnegative martingale. As corollaries, we obtain the strong law of large numbers (SLLN) and the asymptotic equipartition property (AEP) for the bifurcating Markov chains indexed by a binary tree.

Keywords:

• Binary tree
• bifurcating Markov chain
• strong deviation theorem
• asymptotic equipartition property
• strong law of large numbers

MSC (2010):

• 60F15
• 60J10

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (11571142, 11601191), Youth talent cultivation project of Jiangsu University, Young science and technology talents lifting project of Jiangsu association for science and technology.