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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 collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

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new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

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new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

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new correction: cocycle pro cocyle by pahio

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new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier