# Asymptotics and Renewal Approximation in the Online Selection of Increasing Subsequence

## TL;DR

This paper analyzes the online selection of increasing subsequences from a Poisson process, providing detailed asymptotic expansions and a new proof of the central limit theorem using renewal approximation.

## Contribution

It introduces a renewal approximation approach and refines asymptotic estimates for the moments of the subsequence length under optimal strategies.

## Key findings

- Asymptotic expansions for moments of subsequence length
- A novel proof of the central limit theorem using renewal approximation
- Enhanced understanding of the distribution of selected subsequence lengths

## Abstract

We revisit the problem of maximising the expected length of increasing subsequence that can be selected from a marked Poisson process by an online strategy. Resorting to a natural size variable, the problem is represented in terms of a controlled partially deterministic Markov process with decreasing paths. Refining known estimates we obtain fairly complete asymptotic expansions for the moments, and using a renewal approximation give a novel proof of the central limit theorem for the length of selected subsequence under the optimal strategy.