Israel Journal of Mathematics

, Volume 88, Issue 1–3, pp 333–365 | Cite as

Oscillating random walk with a moving boundary

  • Neal Madras
  • David Tanny


A discrete-time Markov chain is defined on the real line as follows: When it is to the left (respectively, right) of the “boundary”, the chain performs a random walk jump with distributionU (respectively,V). The “boundary” is a point moving at a constant speed γ. We examine certain long-term properties and their dependence on γ. For example, if bothU andV drift away from the boundary, then the chain will eventually spend all of its time on one side of the boundary; we show that in the integer-valued case, the probability of ending up on the left side, viewed as a function of γ, is typically discontinuous at every rational number in a certain interval and continuous everywhere else. Another result is that ifU andV are integer-valued and drift toward the boundary, then when viewed from the moving boundary, the chain has a unique invariant distribution, which is absolutely continuous whenever γ is irrational.


Markov Chain Probability Measure Random Walk Invariant Measure Invariant Probability Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    B. R. Bhat,Oscillating random walk and diffusion process under centripetal force, The Indian Journal of Statistics. Series A31 (1969), 391–402.MATHMathSciNetGoogle Scholar
  2. [2]
    A. A. Borovkov,A limit distribution for an oscillating random walk, Theory of Probability and its Applications25 (1980), 649–651.Google Scholar
  3. [3]
    K. L. Chung,A Course in Probability Theory, 2nd ed., Academic Press, New York, 1974.MATHGoogle Scholar
  4. [4]
    K. Golden, S. Goldstein and J. L. Lebowitz,Classical transport in modulated structures, Physical Review Letters55 (1985), 2629–2632.CrossRefMathSciNetGoogle Scholar
  5. [5]
    D. V. Gusak,On oscillating random walk schemes. I, Theory of Probability and Mathematical Statistics39 (1989), 41–46.MATHMathSciNetGoogle Scholar
  6. [6]
    D. V. Gusak,On oscillating random walk schemes. II, Theory of Probability and Mathematical Statistics40 (1990), 11–17.MATHMathSciNetGoogle Scholar
  7. [7]
    P. R. Halmos,Lectures on Ergodic Theory, Chelsea, New York, 1956.MATHGoogle Scholar
  8. [8]
    L. K. Hua,Introduction to Number Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1982.MATHGoogle Scholar
  9. [9]
    J. Keilson and L. D. Servi,Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules, Journal of Applied Probability23 (1986), 790–802.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    J. H. B. Kemperman,The oscillating random walk, Stochastic Processes and their Applications2 (1974), 1–29.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    D. J. Newman,The distribution function for extreme luck, The American Mathematical Monthly67 (1960), 992–994.CrossRefMathSciNetGoogle Scholar
  12. [12]
    S. Orey,Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities, Van Nostrand Reinhold, London, 1971.MATHGoogle Scholar
  13. [13]
    O. E. Percus and J. K. Percus,Piecewise homogeneous random walk with a moving boundary, SIAM Journal on Applied Mathematics47 (1987), 822–831.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    O. E. Percus and J. K. Percus,Coin tossing, revisited, Journal of Applied Probability25 (1988), 70–80.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    D. Revuz,Markov Chains, North-Holland, Amsterdam-Oxford, 1975.MATHGoogle Scholar
  16. [16]
    B. A. Rogozin and S. G. Foss,Recurrency of an oscillating random walk, Theory of Probability and its Applications23 (1978), 155–162.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    M. Rosenblatt,Markov Processes. Structure and Asymptotic Behavior, Springer-Verlag, New York-Berlin-Heidelberg, 1971.MATHGoogle Scholar

Copyright information

© Hebrew University 1994

Authors and Affiliations

  • Neal Madras
    • 1
  • David Tanny
    • 1
  1. 1.Department of Mathematics and StatisticsYork UniversityDownsviewCanada

Personalised recommendations