$Borell-Cantelli Lemma application$

Question

Let $X$ be a uniform random variable on $[0,1]$ and consider the events
$$A_n=\lbrace X\leq \frac{1}{2}+\frac{1}{n}\rbrace, n=1,2,...$$
(a) Compute $\limsup_{n\rightarrow \infty} A_n$.

(I am not great with limsup, so my main question is exacty how to find and justify this answer. My guess is 1/2 but I am not confident.)

(b) Explain why or why not the first Borel-Canteli Lemma can be used to compute $P\lparen \limsup_{n\rightarrow \infty} A_n \rparen=P(A_n i.o.)$ in this example.

(c) Explain why or why not the second Borel-Canteli Lemma can be used to compute $P\lparen \limsup_{n\rightarrow \infty} A_n \rparen=P(A_n i.o.)$ in this example.

Probability

Ian Dumais

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Answer 1

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By definition we have $$ \limsup A_n = \bigcap _{k\ge 1}\bigcup _{n\ge k} A_n = \bigcap _{k\ge 1}\bigcup _{n\ge k}\{ X\le 1/2 +1/n\}=\bigcap _{k\ge 1}\{ X\le 1/2 + 1/k\}=\{X\le 1/2\}$$Note tha the sets $\{X\le a\}$ become larger as $a$ increases, so their union corresponds to the largest $a$ and their intersection corresponds to the smallest $a$.

We also have $\sum _{n\ge 1} P(A_n)=\sum _{n\ge 1} (1/2 +1/n) = \infty$. So the 1st Borel-Canteli Lemma cannot be used since the series is not finite. As for the 2nd lemma, we can use it if $A_n$s form an independent sequence of events. Suppose $n<m$ then $A_m \subset A_n$ and $$ P(A_n \cap A_m)= P(A_m)=1/2 + 1/m \ne (1/2+1/m)(1/2+1/n)=P(A_n)P(A_m)$$So these events are not independent and the 2nd lemma cannot be used either.

Martin

$Borell-Cantelli Lemma application$

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