Probability Theory and Related Fields

, Volume 77, Issue 1, pp 99–116 | Cite as

Stochastic search in a convex region

  • Steven Lalley
  • Herbert Robbins


A stochastic search strategy is proposed for locating a possibility mobile target in a bounded, convex region of the plane. The strategy is asymptotically minimax as ε→0 with respect to the time required to get within ε of the target. The proof involves the study of first passages to time-dependent boundaries by a certain semi-Markov process.


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  1. 1.
    Athreya, K., MacDonald, D., Ney, P.: Limit theorems for semi-Markov processes and renewal theory for Markov chains. Ann. Probab.6, 788–797 (1978)Google Scholar
  2. 2.
    Brown, S.: Optimal search for a moving target in discrete time and space. Oper. Res.28, 1275–1289 (1980)Google Scholar
  3. 3.
    Fitzgerald, C.: The princess and monster differential game. SIAM J. Control Optimization17, 700–712 (1979)Google Scholar
  4. 4.
    Gal, S.: Search games with mobile and immobile hider. SIAM J. Contol Optimization17, 99–122 (1979)Google Scholar
  5. 5.
    Gal, S.: Search games. New York: Academic Press 1980Google Scholar
  6. 6.
    Guillemin, V., Pollack, A.: Differential Tonology. Englewood Cliffs, NJ: Prentice-Hall 1974Google Scholar
  7. 7.
    Isaacs, R.: Differential games. New York: Wiley 1967Google Scholar
  8. 8.
    Kesten, H.: Renewal theory for functionals of a Markov chain with general state space. Ann. Probab.2, 355–386 (1974)Google Scholar
  9. 9.
    Lalley, S., Robbins, H.: Asymptotically minimax stochastic search strategies in the plane. Proc. Natl. Acad. Sci. USA (1987)Google Scholar
  10. 10.
    Lalley, S., Robbins, H.: Stochastic search in a square and on a torus. In: Berger, J., Gupta, S. (eds.) Proc. 4th Purdue Symp. Statist. Dec. Th. 1986Google Scholar
  11. 11.
    Orey, S.: Change of time scale for Markov processes. Trans. Am. Math. Soc.99, 384–390 (1961)Google Scholar
  12. 12.
    Revuz, D.: Markov Chains. Amsterdam: North-Holland 1975Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Steven Lalley
    • 1
  • Herbert Robbins
    • 2
  1. 1.Statistics Department, Math Sciences BuildingPurdue UniversityWest LafayetteUSA
  2. 2.Department of StatisticsRutgers UniversityNew BrunswickUSA

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