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

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 called . -geometric ergodicity and . -geometric recurrence respectively. The first condition generalises geometric as well as strong ergodicity. Our key theorem shows that . -geometric ergodicity is equivalent to weak . -geometric recurrence. The latter condition is verified for the time-discretised two-centre open Jackson network. Hence, the corresponding two-dimensional Markov chain is . -geometrically and geometrically ergodic, but not strongly ergodic. A consequence of . -geometric ergodicity with . of product-form is the convergence of the Laplace-Stieltjes transforms of the marginal distributions. Consequently all moments converge.