Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Ergodicity conditions for two-dimensional Markov chains on the positive quadrant
Download PDF
Download PDF
  • Published: September 1989

Ergodicity conditions for two-dimensional Markov chains on the positive quadrant

  • Walter A. Rosenkrantz1 

Probability Theory and Related Fields volume 83, pages 309–319 (1989)Cite this article

  • 178 Accesses

  • 9 Citations

  • Metrics details

Summary

The basic problem considered in this paper is that of determining conditions for recurrence and transience for two dimensional irreducible Markov chains whose state space is Z 2+ =Z+xZ+. Assuming bounded jumps and a homogeneity condition Malyshev [7] obtained necessary and sufficient conditions for recurrence and transience of two dimensional random walks on the positive quadrant. Unfortunately, his hypothesis that the jumps of the Markov chain be bounded rules out for example, the Poisson arrival process. In this paper we generalise Malyshev's theorem by means of a method that makes novel use of the solution to Laplace's equation in the first quadrant satisfying an oblique derivative condition on the boundaries. This method, which allows one to replace the very restrictive boundedness condition by a moment condition and a lower boundedness condition, is of independent interest.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Baccelli, F.: Exponential martingales and Wald's formulas for two queue networks. J. Appl. Probab.23,812–819 (1986)

    Google Scholar 

  2. Fayolle, G., King, P.J.B., Mitrani, I.: The solution of certain two dimensional Markov Models. Adv. Appl. Probab.14, 295–308 (1982)

    Google Scholar 

  3. Foster, F.G.: On the Stochastic Matrices associated with certain queueing processes. Ann. Math. Stat.24, 355–360 (1953)

    Google Scholar 

  4. Harrison, J.M., Williams, R.J.: Brownian models of open queueing networks with homogeneous customer populations. Stochastics22, 77–115 (1987)

    Google Scholar 

  5. Kingman, J.F.C.: The ergodic behaviour of random walks. Biometrika48, 391–396 (1961)

    Google Scholar 

  6. Lamperti, J.: Criteria for the recurrence or transience of stochastic processes. I. J. Math. Anal. Appl.1, 314–330 (1960)

    Google Scholar 

  7. Malysev, V.A.: Classification of two-dimensional positive random walks and almost linear semimartingales. Sov. Math., Dokl.13, 136–139 (1972)

    Google Scholar 

  8. Varadhan, S.R.S., Williams, R.: Brownian motion in a wedge with oblique reflection. Commun. Pure Appl. Math,38, 405–443 (1985)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, 01003, Amherst, MA, USA

    Walter A. Rosenkrantz

Authors
  1. Walter A. Rosenkrantz
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rosenkrantz, W.A. Ergodicity conditions for two-dimensional Markov chains on the positive quadrant. Probab. Th. Rel. Fields 83, 309–319 (1989). https://doi.org/10.1007/BF00964367

Download citation

  • Received: 19 October 1987

  • Revised: 07 March 1988

  • Issue Date: September 1989

  • DOI: https://doi.org/10.1007/BF00964367

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Markov Chain
  • Mathematical Biology
  • Basic Problem
  • Moment Condition
  • Arrival Process
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature