Refined Asymptotics in the Online Selection of an Increasing Subsequence

TL;DR

This paper provides a refined asymptotic expansion for the maximum expected length of an increasing subsequence selected online, using dynamic programming analysis, applicable to both fixed sample sizes and Poisson process observations.

Contribution

It introduces a detailed asymptotic expansion for the online selection problem, improving previous estimates with a new analytical approach.

Findings

Asymptotic expansion of $v_n$ up to an $O(1)$ term.

Method applicable to Poisson process observation times.

Enhanced understanding of online increasing subsequence selection.

Abstract

Let $v_n$ be the maximum expected length of an increasing subsequence, which can be selected by an online nonanticipating policy from a random sample of size $n$. Refining known estimates, we obtain an asymptotic expansion of $v_n$ up to a $O(1)$ term. The method we use is based on detailed analysis of the dynamic programming equation, and is also applicable to the online selection problem with observations occurring at times of a Poisson process.