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# 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.

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

60A99*no label found*

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new correction: Define Galois correspondence by porton

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new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

new question: Lorenz system by David Bankom

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new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

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new question: Latent variable by adam_reith