On the Borel-Cantelli Lemma and its Generalization

TL;DR

This paper presents a generalized version of the Borel-Cantelli lemma, providing a lower bound for the probability of infinitely many events occurring based on weighted sums and their intersections.

Contribution

It introduces a new inequality that extends the classical Borel-Cantelli lemma by incorporating weights and intersection probabilities.

Findings

Provides a lower bound for P(limsup A_n) using weighted sums.

Generalizes the Borel-Cantelli lemma with a new inequality.

Applicable to sequences of events with weighted probabilities.

Abstract

Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of events on a probability space $(\Omega,\mathcal{F},\mathbf{P})$. We show that if $\lim_{m\to\infty}\sum_{n=1}^{m}w_n\mathbf{P}(A_n)=\infty$ where each $w_n\in\mathbb{R}$, then \[{\mathbf{P}}(\limsup A_n)\geq\limsup_{n\to\infty} \frac{\displaystyle\big(\sum_{k=1}^n{w_k\mathbf{P}}(A_k)\big)^2}{\displaystyle\sum_{i=1}^n\sum_{j=1}^nw_iw_j{\mathbf{P}}(A_i\cap A_j)}.\]