BorelCantelli lemma
Let ${A}_{1},{A}_{2},\mathrm{\dots}$ be random events in a probability space^{}.

1.
If $$, then $P({A}_{n}\mathrm{i}.\mathrm{o}.)=0$;

2.
If ${A}_{1},{A}_{2},\mathrm{\dots}$ are independent^{}, and ${\sum}_{n=1}^{\mathrm{\infty}}P({A}_{n})=\mathrm{\infty}$, then $P({A}_{n}\mathrm{i}.\mathrm{o}.)=1$
where $A=[{A}_{n}\mathrm{i}.\mathrm{o}.]$ represents the event “${A}_{n}$ happens for infinitely many values of $n$.” Formally, $A=lim\; sup{A}_{n}$, which is a limit superior of sets.
Title  BorelCantelli lemma^{} 

Canonical name  BorelCantelliLemma 
Date of creation  20130322 13:13:18 
Last modified on  20130322 13:13:18 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  7 
Author  Koro (127) 
Entry type  Theorem 
Classification  msc 60A99 