An Optional Stopping-like theorem for large stopping times on birth-death chains

TL;DR

This paper provides an elementary proof of a theorem demonstrating that certain birth-death chains exhibit martingale-like behavior at large stopping times, extending previous results with a new proof approach.

Contribution

It introduces a simplified proof for a generalized theorem on birth-death chains' behavior at large stopping times, expanding on prior work.

Findings

Birth-death chains show martingale-like behavior at large stopping times

Elementary proof simplifies understanding of the theorem

Generalizes previous results on stopping times

Abstract

An elementary proof is given for a theorem showing that certain birth-death chains show martingale-like behavior at large stopping times. This is a generalization of and new proof for a theorem from a earlier paper by the author.