Optimal Stopping for the Uniform Distribution

Many discrete-time optimal stopping problems are known to have more tractable limit forms based on a planar Poisson process. Using this tool we find a solution to the optimal stopping problem for i.i.d. sequence of $n$ discrete uniform random variables, in the asymptotic regime where $n$ and the range of distribution are of the same order. The optimal stopping rule in the Poisson problem is identified, by means of a time change, with known asymptotic solution to Lindley's problem of minimising the expected rank.