On the Borel-Cantelli Lemma and its Generalization

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 \[{\mathbf{P}}(\limsup A_n)\geq\limsup_{n\to\infty} \frac{\displaystyle\big(\sum_{k=1}^n{w_k\mathbf{P}}(A_k)\big)^2}{\displaystyle\sum_{i=1}^n\sum_{j=1}^nw_iw_j{\mathbf{P}}(A_i\cap A_j)}.\]

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