Optimal Online Selection of a Monotone Subsequence: a Central Limit Theorem

TL;DR

This paper establishes a central limit theorem for the optimal number of selections in an online monotone subsequence problem, providing insights into its mean and variance for large sequences.

Contribution

It introduces a central limit theorem for the distribution of the optimal selection count in an online setting, extending previous finite-sequence results.

Findings

Central limit theorem for $L_n(\pi_n^*)$

Asymptotic mean and variance characterization

Comparison with finite sequence Poisson models

Abstract

Consider a sequence of $n$ independent random variables with a common continuous distribution $F$, and consider the task of choosing an increasing subsequence where the observations are revealed sequentially and where an observation must be accepted or rejected when it is first revealed. There is a unique selection policy $\pi_n^*$ that is optimal in the sense that it maximizes the expected value of $L_n(\pi_n^*)$, the number of selected observations. We investigate the distribution of $L_n(\pi_n^*)$; in particular, we obtain a central limit theorem for $L_n(\pi_n^*)$ and a detailed understanding of its mean and variance for large $n$. Our results and methods are complementary to the work of Bruss and Delbaen (2004) where an analogous central limit theorem is found for monotone increasing selections from a finite sequence with cardinality $N$ where $N$ is a Poisson random variable that is independent of the sequence.