On the Borel-Cantelli Lemmas, the Erd\H{o}s-R\'enyi Theorem, and the Kochen-Stone Theorem

TL;DR

This paper provides a detailed quantitative analysis of classical probability lemmas and theorems, revealing explicit relationships and computability issues, and introduces a metastability-based formulation for the Kochen-Stone Theorem.

Contribution

It offers the first explicit quantitative formulations of the Borel-Cantelli lemmas and Erdős-Rényi theorem, and introduces a metastability-based approach for the Kochen-Stone theorem.

Findings

Explicit quantitative bounds for Borel-Cantelli and Erdős-Rényi theorems.

Non-computability of direct bounds for the Kochen-Stone theorem.

Quantitative formulation of Kochen-Stone using Tao's metastability.

Abstract

In this paper we present a quantitative analysis of the first and second Borel-Cantelli Lemmas and of two of their generalisations: the Erd\H{o}s-R\'enyi Theorem, and the Kochen-Stone Theorem. We will see that the first three results have direct quantitative formulations, giving an explicit relationship between quantitative formulations of the assumptions and the conclusion. For the Kochen-Stone theorem, however, we can show that the numerical bounds of a direct quantitative formulation are not computable in general. Nonetheless, we obtain a quantitative formulation of the Kochen-Stone Theorem using Tao's notion of metastability.