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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

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Reference

## Mathematics Subject Classification

60A99*no label found*

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