On sequential selection and a first passage problem for the Poisson process

TL;DR

This paper explores the connection between online and offline subsequence selection problems in Poisson processes, providing asymptotic results for counting arrivals before a sum constraint is exceeded, using coupling techniques.

Contribution

It introduces a novel analysis linking subsequence selection in Poisson processes with a first passage problem, deriving precise asymptotics for the mean count.

Findings

Asymptotic formulas for the mean count of Poisson arrivals before sum exceeds a level.

Coupling with a nonlinear pure birth process to analyze the problem.

Connections established between online and offline subsequence selection problems.

Abstract

This note is motivated by connections between the online and offline problems of selecting a possibly long subsequence from a Poisson-paced sequence of uniform marks under either a monotonicity or a sum constraint. The offline problem with the sum constraint amounts to counting the Poisson arrivals before their total exceeds a certain level. A precise asymptotics for the mean count is obtained by coupling with a nonlinear pure birth process.