Journal of Statistical Physics

, Volume 30, Issue 3, pp 755–783 | Cite as

Phase transition in a four-dimensional random walk with application to medical statistics

  • O. E. Percus
  • J. K. Percus


A random walk in a piecewise homogeneous medium can exhibit a variety of asymptotic behaviors. In particular, it may lodge strictly in one region or divide in probability among several. This will depend upon the parameters describing (a) the walk, (b) the interregion boundary, and (c) the initial location of the walk. We analyze from this point of view a special four-dimensional walk on an integer lattice with two homogeneous regions separated by a hyperplane of codimension 1. The walk represents a continuing sequence of clinical trials of two drugs of unknown success probabilities and the two regions represent the Bayes-derived criterion as to which drug to try next. The demarcation in the parameter space of success probabilities and initial coordinates between one- and two-region asymptotics is mapped out analytically in several special cases and supporting numerical evidence given in the general case.

Key words

Phase transition random walk integer lattice clinical trials piecewise homogeneous 


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  1. 1.
    W. Feller,An Introduction to Probability Theory and its Applications, Vol. I (John Wiley, New York, 1968).Google Scholar
  2. 2.
    Symposium on Random Walks, NBS, June 1982, to be published inJ. Stat. Phys. 30 (1983).Google Scholar
  3. 3.
    H. E. Robbins, “Some aspects of sequential design of experiments,”Bull. Am. Math. Soc. 58:527–535 (1952).Google Scholar
  4. 4.
    O. E. Percus and J. K. Percus, “Ethical procedures in comparative evaluation of drugs,” Courant Mathematics and Computing Laboratory Rep., New York University, New York (1980).Google Scholar
  5. 5.
    J. A. Bather, “Randomized allocation of treatment in sequential experiments,”J. Royal Stat. Soc. (B) 265–292 (1981).Google Scholar
  6. 6.
    Y. S. Chow, H. E. Robbins, and D. Siegmund,Great Expectation: The Theory of Optimal Stopping (Houghton-Mifflin, New York, 1971).Google Scholar
  7. 7.
    M. L. Aggarwal, “Records and oscillating random walk,”Calcutta Stat. Assoc. Bull. 25:55–64 (1977).Google Scholar
  8. 8.
    O. E. Percus and J. K. Percus, “One dimensional random walk with phase transition,”SIAM J. Appl. Math. 40:485–497 (1981).Google Scholar
  9. 9.
    For numerical work, see R. E. Bechhofer and R. V. Kulkarni, “Closed adaptive sequential procedures for selecting the best ofk ⩾ 2 Bernoulli populations,” College of Engineering, Cornell Univ. Report (1981).Google Scholar
  10. 10.
    For analytical treatment, see O. E. Percus and J. K. Percus, to be published.Google Scholar
  11. 11.
    E. T. Whittaker and G. N. Watson,A Course of Modern Analysis, (Cambridge Univ. Press, Cambridge, 1962).Google Scholar

Copyright information

© Plenum Publishing Corporation 1983

Authors and Affiliations

  • O. E. Percus
    • 1
  • J. K. Percus
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew York

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