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Searching for a one-dimensional random walker

Published online by Cambridge University Press:  14 July 2016

Bernard J. McCabe
Bernard J. McCabe*
Affiliation:
Daniel H. Wagner, Associates, Paoli, Pennsylvania
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Abstract

Let {xk}k ≧ − r be a simple Bernoulli random walk with x–r = 0. An integer valued threshold ϕ = {ϕk}k≧1 is called a search plan if |ϕk+1−ϕk|≦1 for all k ≧ 1. If ϕ is a search plan let τϕ be the smallest integer k such that x and ϕ cross or touch at k. We show the existence of a search plan ϕ such that ϕ1 = 0, the definition of ϕ does not depend on r, and the first crossing time τϕ has finite mean (and in fact finite moments of all orders). The analogous problem for the Wiener process is also solved.

Keywords

SEARCH THEORYTHRESHOLD CROSSINGRANDOM WALK
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 1 , March 1974 , pp. 86 - 93
Copyright
Copyright © Applied Probability Trust 1974 

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References

Beck, A. (1964) On the linear search problem Israel J. Math. 2, 221–228.Google Scholar
Cramer, H. (1946) Mathematical Methods of Statistics. Princeton University Press, Princeton, N. J. Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. Wiley, New York.Google Scholar