Borel-Cantelli Lemma

Lemma ( Fatou's Lemma ) (Fatou's Lemma) [1]

$$P(\liminf A_n) \leq \liminf P(A_n) \qquad \limsup P(A_n) \leq P(\limsup A_n)$$

Proof) 

$B_k =\cap_{n\geq k}A_n$ 은 increasing sequence of event이다. 

또한 $B_k$의 정의로부터 $B_k \subset A_n$ for any $n$ such that $n\geq k$ 임을 알 수 있다.

즉 $P(B_k) \leq \inf_{n\geq k}P(A_n)$이다.

따라서 

$P(\liminf_n A_n) = P( \cup_n B_b) = \lim P(B_n)$ 이다.

여기서 $B_n$이 increasing sequence라는 점을 활용했다.

결론적으로, $P(\liminf_n A_n) = \lim_n P (B_n) \leq \liminf_n P(A_n)$이 성립한다.

Q.E.D

 

Theorem ( Borel Cantelli ) 

If $ \sum\limits^{\infty}_{i=1} P(A_n) < \infty $, then $P(\limsup A_n)=0$

Proof)

Let $N= \sum\limits^{\infty}_1 1_{A_n}$

$w\in \limsup A_n$ is equivalent to $N=\infty$

$EN = E \sum\limits 1_{A_n} = \sum\limits E 1A_n = \sum\limits P (A_n) < \infty $

Therefore we have $N < \infty$ a.s.

 

Theorem 2.3.8 

If $X_1,X_2,\dots$ are i.i.d with $E|X_i| = \infty$ then $P( |X_n| \geq n \;i.o)=1$. So if $S_n = X_1 + \dots X_n$ then $P(\lim S_n /n \; \text{exists}\in (-\infty,\infty))=0$

Since $ E|X_1| = \infty$,

$E f(x) = \int^{\infty}_0 f'(x) P ( X\geq x) dx$

$E |X| = \int^{\infty}_0  P ( |X| > x) dx \leq \sum\limits^{\infty}_{i=0} P (|X_1|>n) $

Hence $P(|X_n| \geq n \; i.o)=1$

Now observe that

$$\frac{S_n}{n}-\frac{S_{n+1}}{n+1}= \frac{S_n}{n(n+1)} - \frac{X_{n+1}}{n+1}$$ 

Assume $\lim \frac{S_n}{n}$ exists. Then \frac{X_{n+1}}{n+1}\rightarrow 0$ which contradicts $P(|X_n| \geq n \; i.o)=1$

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References

[1] https://www.math.bgu.ac.il/~yadina/teaching/probability/prob2.pdf