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1 results

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

  • Part of
  • Arie Hordijk, Flora Spieksma
  • Journal:
    Advances in Applied Probability / Volume 24 / Issue 2 / June 1992
    Published online by Cambridge University Press:
    01 July 2016, pp. 343-376
    Print publication:
    June 1992

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