On ergodicity and recurrence properties of a Markov chain by an application to an open jackson network

Abstract

This paper gives an overview of recurrence and ergodicity properties of a Markov chain. Two new notions for ergodicity and recurrence are introduced. They are calledμ-geometric ergodicity andμ-geometric recurrence respectively. The first condition generalises geometric as well as strong ergodicity. Our key theorem shows thatμ-geometric ergodicity is equivalent to weakμ-geometric recurrence. The latter condition is verified for the time-discretised two-centre open Jackson network. Hence, the corresponding two-dimensional Markov chain isμ-geometrically and geometrically ergodic, but not strongly ergodic. A consequence ofμ-geometric ergodicity withμof product-form is the convergence of the Laplace-Stieltjes transforms of the marginal distributions. Consequently all moments converge.