On the ergodic decomposition for a class of Markov chains

Author(s)

Costa, O. L. V. ; Dufour, F.

• Document type: Journal article
• Source: Stochastic Processes and their Applications; v. 115, n. 3, p. 401-415, Mar. 2005.
• Field of knowledge: Engineering - Electrical Engineering

Abstract

In this paper we present sufficient conditions for the Doeblin decomposition, and necessary and sufficient conditions for an ergodic decomposition for a Markov chain satisfying a T'-condition, which is a condition adapted from the paper (Statist. and Probab. Lett. 50 (2000) 13). Under no separability assumption on the sigma-field, it is shown that the T'-condition is sufficient for the condition that there are no uncountable disjoint absorbing sets and, under some hypothesis, it is also necessary. For the case in which the sigma-field is countable generated and separated, this condition is equivalent to the existence of a T continuous component for the Markov chain. Furthermore, under the assumption that the space is a compact separable metric space, it is shown that the Foster-Lyapunov criterion is necessary and sufficient for the existence of an invariant probability measure for the Markov chain, and that every probability measure for the Markov chain is, in this case, non-singular. (AU)

• FAPESP's process: 03/06736-7 - Control and filtering of Markovian jumping parameters stochastic systems
• Grantee: João Bosco Ribeiro do Val
• Support Opportunities: Research Projects - Thematic Grants