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-McMillan theorem for a finite second-order Markov chain indexed by this tree are obtained.2000 Mathematics Subject Classification: 60F15; 60J10.

This paper presents a convergence theorem for a finite second-order Markov chain indexed by a general infinite tree with uniformly bounded degree, and obtains the strong law of large numbers and Shannon-McMillan theorem for a finite second-order Markov chain

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

Key Takeaway: This paper presents a convergence theorem for a finite second-order Markov chain indexed by a general infinite tree with uniformly bounded degree, and obtains the strong law of large numbers and Shannon-McMillan theorem for a finite second-order Markov chain