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Borel-Cantelli lemma
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(Theorem)
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Let $A_1, A_2,\dots$ be random events in a probability space.
- If $\sum_{n=1}^\infty P(A_n)<\infty$ then $P(A_n \operatorname{i.o.}) = 0$
- 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.
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"Borel-Cantelli lemma" is owned by Koro.
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Cross-references: limit superior of sets, represents, independent, probability space, random events
There are 2 references to this entry.
This is version 4 of Borel-Cantelli lemma, born on 2002-12-08, modified 2004-02-23.
Object id is 3688, canonical name is BorelCantelliLemma.
Accessed 16053 times total.
Classification:
| AMS MSC: | 60A99 (Probability theory and stochastic processes :: Foundations of probability theory :: Miscellaneous) |
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Pending Errata and Addenda
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