On optimal search plans to detect a target moving randomly on the real line
type: Item
instance of: scholarly article
title: On optimal search plans to detect a target moving randomly on the real line
zbMATH Open document ID: 0574.60076
DOI: 10.1016/0304-4149(85)90218-2
author: John G. Wilson
published in: Stochastic Processes and their Applications
publication date: 1985-01-01T00:00:00Z
review text: A target is performing a random walk \(\{X_ n\}^{\infty}_{n=0}\) on the integers, with probability p of moving away from the origin and 1-p of moving towards the origin at each step, where \(0<p<1\). The distribution of \(x_ 0\) is assumed known. A searcher is assumed to follow a deterministic search plan \(\xi =\{\xi_ n\}^{\infty}_{n=0}\), where \(\xi_ n\) (an integer) is the position at time n. Let \(\tau_{\xi}\) denote the time of capture, more precisely \(\tau_{\xi}=\min \{n:(x_ n-\xi_ n)(x_{n-1}-\xi_{n-1})\leq 0\}.\) The main result of the paper is that for any \(k\geq 1\), a necessary and sufficient condition for the existence of a search plan \(\xi\) with \(E[\tau^ k_{\xi}]<\infty\) is that \(E[| x^ k_ 0|]<\infty\). Moreover, it is shown that in this case there exists a search plan \(\xi\) minimizing \(E[\tau^ k_{\xi}]\). The following plan is shown to have finite \(E[\tau^ k_{\xi}]\). For some \(\alpha \geq \max [200(1-p)^{- 3}\), \(200p^{-3}]\), start at 0, move to 1, then to -\(\alpha\), then to \(\alpha^ 2\), then to \(-\alpha^ 3\), etc.
Mathematics Subject Classification ID: 60J20
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 3917397
zbMATH Keywords: Fibonacci sequence
zbMATH Keywords: search theory
zbMATH Keywords: deterministic search plan
MaRDI profile type: Publication
full work available at URL: 0304-4149(85)90218-2
OpenAlex ID: W2032500333
cites work: Optimal Search for a Moving Target in Discrete Time and Space
cites work: Q5538132
cites work: Searching for a one-dimensional random walker
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description: scientific article; zbMATH DE number 3917397