On Markov chains induced by partitioned transition probability matrices

TL;DR

This paper studies the convergence behavior of Markov chains generated by partitions of transition probability matrices, with applications to filtering in partially observed Markov processes.

Contribution

It introduces a framework for analyzing Markov chains induced by matrix partitions and establishes convergence results relevant to filtering in partially observed systems.

Findings

Established conditions for convergence in distribution of these Markov chains.

Applied results to filtering processes in partially observed Markov chains.

Provided theoretical insights into the structure of partition-induced Markov chains.

Abstract

Let S be a denumerable state space and let P be a transition probability matrix on S. If a denumerable set M of nonnegative matrices is such that the sum of the matrices is equal to P, then we call M a partition of P. Let K denote the set of probability vectors on S. To every partition M of P we can associate a transition probability function on K defined in such a way that if p in K and m in M are such that ||pm|| > 0, then, with probability ||pm|| the vector p is transferred to the vector pm/||pm||. Here ||.|| denotes the l_1-norm. In this paper we investigate convergence in distribution for Markov chains generated by transition probability functions induced by partitions of transition probability matrices. An important application of the convergence results obtained is to filtering processes of partially observed Markov chains.