Optimal Stopping for the Uniform Distribution

TL;DR

This paper addresses the optimal stopping problem for sequences of discrete uniform variables by leveraging Poisson process limits, providing an asymptotic solution that connects to Lindley's problem of minimizing expected rank.

Contribution

It introduces a novel approach using planar Poisson processes to solve the uniform distribution stopping problem in the asymptotic regime.

Findings

Derived the asymptotic optimal stopping rule for uniform variables

Connected the solution to Lindley's problem of minimizing expected rank

Established a limit form using planar Poisson processes

Abstract

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.