On optimal search plans to detect a target moving randomly on the real line
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.
scientific article; zbMATH DE number 3917397
scientific article