A Note on the Borel-Cantelli Lemma

TL;DR

This paper examines the Barndorff-Nielsen lemma, clarifies its proof, offers an alternative demonstration, and introduces a new generalization of the classical Borel-Cantelli lemma.

Contribution

It corrects and clarifies the proof of the Barndorff-Nielsen lemma and proposes a novel generalization of the Borel-Cantelli lemma.

Findings

The original proof of the Barndorff-Nielsen lemma is incorrect.

An alternative proof of the lemma is provided.

A new generalization of the Borel-Cantelli lemma is introduced.

Abstract

In this short note, we discuss the Barndorff-Nielsen lemma, which is a generalization of well-known Borel-Cantelli lemma. Although the result stated in the Barndorff-Nielsen lemma is correct, it does not follow from the argument proposed in the corresponding proof. In this note, we show this and offer an alternative proof of this lemma. We also propose a new generalization of Borel-Cantelli lemma.