Question

Problem 1.25. Suppose you are given a sequence of events An, nEN that are independent and such that ΣηΕΝ P(An-oo. Prove that the event "An happens infinitely often" has probability one. This result is the reverse iel Note it nes the events to he independent.

Answer 1

$\sum_{n\in N}^{\,}P(A_{n})=\infty$

Let A_n' denote the compliment of A_n.

Now; P(A_n happens infinitely often)

$=P\left ( \bigcup_{n\in N}^{\,}A_{n} \right ) =1-P\left [ \left ( \bigcup_{n\in N}^{\,}A_{n} \right )' \right ] =1-P\left ( \bigcap_{n\in N}^{\,}A_{n}' \right )$ [according to De-Morgan's law]

Now; A_n are independent $\forall$ n$\in$N $\Rightarrow$ A_n' are independent $\forall$ n$\in$N

$\therefore P\left ( \bigcap_{n\in N}^{\,}A_{n}' \right )=\prod_{n\in N}^{\,}P(A_{n}')=\prod_{n\in N}^{\,}[1-P(A_{n})]$

For a real number x$\in$(0,1) ; 1-x$\leq$e^-x

$\therefore$  P(A_n)$\in$(0,1) $\forall$ n$\in$N $\Rightarrow$ 1-P(A_n)$\leq$e^-P(A_n)$\forall$ n$\in$N

  $\Rightarrow \prod_{n\in N}^{\,}[1-P(A_{n})]\leq \prod_{n\in N}^{\,}e^{-P(A_{n})}=e^{-\sum_{n\in N}^{\,}P(A_{n})}=e^{-\infty }=0$

$\Rightarrow P\left ( \bigcap_{n\in N}^{\,}A_{n}' \right )\leq 0$

$P\left ( \bigcap_{n\in N}^{\,}A_{n}' \right )\;cannot\;be\;less\;than\;0$   $\therefore P\left ( \bigcap_{n\in N}^{\,}A_{n}' \right )=0$  

$\therefore$ P(A_n happens infinitely often)

$\therefore P(A_{n}\;\,happens\;\,infinitely\;\,often)=1-P\left ( \bigcap_{n\in N}^{\,}A_{n}' \right )=1-0=1$

PROVED