Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T22:36:02.218Z Has data issue: false hasContentIssue false
  • English
  • Français

DISCRETE SCATTERING AND SIMPLE AND NONSIMPLE FACE-HOMOGENEOUS RANDOM WALKS

Published online by Cambridge University Press:  19 March 2008

Arie Hordijk ,
Nikolay Popov  and
Flora Spieksma
Arie Hordijk
Affiliation:
Mathematics Institute University of Leiden2300RA Leiden, The Netherlands E-mail: [email protected]; [email protected]
Nikolay Popov
Affiliation:
Mathematics Institute University of Leiden2300RA Leiden, The Netherlands E-mail: [email protected]; [email protected]
Flora Spieksma
Affiliation:
Mathematics Institute University of Leiden2300RA Leiden, The Netherlands E-mail: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article we will derive some results for characterizing the almost closed sets of a face-homogeneous random walk. We will present a conjecture on the relation between discrete scattering of the fluid limit and the absence of nonatomic almost closed sets. We will illustrate the conjecture with random walks with both simple and nonsimple decomposition into almost closed sets.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 22 , Issue 2 , March 2008 , pp. 163 - 189
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Blackwell, D. (1955). On transient Markov processes with a countable number of states and stationary transition probabilities. Annals of Mathematical Statistics 26: 654–658.CrossRefGoogle Scholar
2.Chung, K.L. (1960). Markov chains with stationary transition probabilities. Berlin: Springer-Verlag.CrossRefGoogle Scholar
3.Fayolle, G., Malyshev, V.A. & Menshikov, M.V. (1995). Constructive theory of countable Markov chains. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
4.Feller, W. (1956). Boundaries induced by non-negative matrices. Transactions of the American Mathematical Society 83: 19–54.CrossRefGoogle Scholar
5.Hall, P. & Heyde, C.C. (1980). Martingale limit theory and its application. San Diego: Academic Press.Google Scholar
6.Hordijk, A. & Popov, N.V. (2003). Large deviation bounds for face-homogeneous random walks in the quarter plane. Probability in the Engineering and Informational Sciences 17: 369–395.CrossRefGoogle Scholar
7.Hordijk, A. & Popov, N.V. (2003). Large deviation analysis for a coupled processors system. Probability in the Engineering and Informational Sciences 17: 397–409.CrossRefGoogle Scholar
8.Kaimanovich, V.A. (1992). Measure-theoretic boundaries of Markov chains, 0–2 laws and entropy. In Picardello, M. A. (ed.), Harmonic analysis and discrete potential theory, Frascati, 1991. New York: Plenum, 145–180.CrossRefGoogle Scholar
9.Kurkova, I.A. (1999). The Poisson boundary for homogeneous random walks. Russian Mathematical Surveys 54: 441–442.CrossRefGoogle Scholar
10.Popov, N. & Spieksma, F.M. (2002). Non-existence of a stochastic fluid limit for a cycling random walk. Technical report, Mathematical Institute, University of Leiden.Google Scholar
11.Ross, S.M. (1970). Applied probability models with optimization applications. San Francisco: Holden Day.Google Scholar
12.Spieksma, F.M. (2008). Continuous scattering and non-atomicity of a face-homogeneous random walk. In preparation.Google Scholar
13.Spieksma, F.M. (2007). Lyapunov functions for Markov chains with applications to face-homogeneous random walks. Internal communication.Google Scholar
14.Spitzer, F.L. (1976). Principles of random walk, 2nd ed.Berlin: Springer-Verlag.CrossRefGoogle Scholar
15.Williams, D. (1991). Probability with martingales. Cambridge: Cambridge University Press.CrossRefGoogle Scholar