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Optimal Stopping for the Uniform Distribution

Alexander Gnedin

TL;DR

The work studies optimal stopping for i.i.d. uniform draws, first revisiting Moser’s problem and deriving its planar Poisson limit to obtain the asymptotic rule $v(T)=2/T$. It then transfers to the discrete-uniform setting with horizon $n$ and range $N$ in the regime $n/N\to T$, where a Poisson limit yields the limit $u(T)$ and a family of beta strategies with cutoffs $\delta_k$. A key contribution is showing that, via a simple time change, Lindley’s minimum-rank problem emerges with a fixed limit $R_n\to 3.869...$, while the discrete-uniform problem yields explicit two-sided bounds and the infinite-product structure for the cutoff sequence. The results connect three classic stopping problems through the planar Poisson framework and provide explicit asymptotic strategies, including the optimal $b=2$ beta-strategy, together with detailed cutoff recursions and densities. The findings offer a principled bridge between discrete and continuous stopping problems and advance the toolbox for Poisson-based optimal stopping with rank- and value-based losses.

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.

Optimal Stopping for the Uniform Distribution

TL;DR

The work studies optimal stopping for i.i.d. uniform draws, first revisiting Moser’s problem and deriving its planar Poisson limit to obtain the asymptotic rule . It then transfers to the discrete-uniform setting with horizon and range in the regime , where a Poisson limit yields the limit and a family of beta strategies with cutoffs . A key contribution is showing that, via a simple time change, Lindley’s minimum-rank problem emerges with a fixed limit , while the discrete-uniform problem yields explicit two-sided bounds and the infinite-product structure for the cutoff sequence. The results connect three classic stopping problems through the planar Poisson framework and provide explicit asymptotic strategies, including the optimal beta-strategy, together with detailed cutoff recursions and densities. The findings offer a principled bridge between discrete and continuous stopping problems and advance the toolbox for Poisson-based optimal stopping with rank- and value-based losses.

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 discrete uniform random variables, in the asymptotic regime where 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.
Paper Structure (17 sections, 5 theorems, 70 equations, 2 figures)

This paper contains 17 sections, 5 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Moser's problem and its limit form
  3. Moser's problem in a nutshell
  4. The planar Poisson framework
  5. Variational principles
  6. Dynamic programming
  7. The calculus of variations approach
  8. Selfsimilarity
  9. Details on beta strategies
  10. Sampling from the discrete uniform distribution
  11. The discrete time stopping problem
  12. The Poisson counterpart
  13. Example
  14. Lindley's minimum rank problem
  15. Lower bound
  16. ...and 2 more sections

Key Result

Theorem 1

The stopping value and the optimal stopping time for problem (POS) are given by, respectively,

Figures (2)

  • Figure 1: For sampling from $\{0, 1,\ldots, N-1\}$ (respectively, from $\{1,2,\ldots,N\}$), a draw falling strictly (respectively, nonstrictly) below the step curve must be accepted when $n\approx TN$ trials remain. The lower smooth curve is the limit stopping value $u(T)$ for the discrete-uniform problem, and the upper smooth curve is the limit $v(T)=2/T$ for Moser's problem.
  • Figure 2: For $T\approx -\log(j/n)$ the relative rank $Y_j$ falling strictly below the step curve must be accepted. The smooth curves are $v(e^{-T}) <w(T)<h(e^{-T})$.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Theorem 4