Searching for a one-dimensional random walker

Abstract

Let {x_k }_{k ≧ − r} be a simple Bernoulli random walk with x _–r = 0. An integer valued threshold ϕ = {ϕ_k }_k≧1 is called a search plan if |ϕ_k+1−ϕ_k |≦1 for all k ≧ 1. If ϕ is a search plan let τ_ϕ 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 ϕ ₁ = 0, the definition of ϕ does not depend on r, and the first crossing time τ_ϕ has finite mean (and in fact finite moments of all orders). The analogous problem for the Wiener process is also solved.