Borel-Cantelli lemma

Let $A_{1},A_{2},\dots$ be random events in a probability space.

1.

If $\sum_{n=1}^{\infty}P(A_{n})<\infty$ , then $P(A_{n}\operatorname{i.o.})=0$ ;
2.

If $A_{1},A_{2},\dots$ are independent, and $\sum_{n=1}^{\infty}P(A_{n})=\infty$ , then $P(A_{n}\operatorname{i.o.})=1$

where $A=[A_{n}\operatorname{i.o.}]$ represents the event “ $A_{n}$ happens for infinitely many values of $n$ .” Formally, $A=\limsup A_{n}$ , which is a limit superior of sets.

Title	Borel-Cantelli lemma
Canonical name	BorelCantelliLemma
Date of creation	2013-03-22 13:13:18
Last modified on	2013-03-22 13:13:18
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	7
Author	Koro (127)
Entry type	Theorem
Classification	msc 60A99