Strong Law of Large Numbers for Branching Markov Chains

C. Takacs

2002, v.8, Issue 1, 107-116

ABSTRACT

We consider the Markov chains whose index-sets are infinite trees. Given a certain state, we calculate its relative frequency up to generation $n$ of the tree and consider the limit $n \rightarrow \infty$. In this setting we prove a strong law of large numbers for Markov chains with finite state space and irreducible aperiodic transition matrix, and trees with uniformly bounded degree.

Keywords: strong law of large numbers,infinite tree,branching Markov chain,population composition