1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

P[limsup] of independent events

  1. Aug 14, 2009 #1
    1. The problem statement, all variables and given/known data

    Let (A_n) be a sequence of independent events such that Pr(A_n)<1 for all n. Show that P[limsup A_n]=1 if and only if P(\cup_{n=1}^{+\infty} A_n)=1

    2. Relevant equations



    3. The attempt at a solution
    Suppose P[\limsup A_n]=1. Define $B_k=\cup_{i=k}^{+\infty} A_i so that B_n\downarrow \limsup_{n\to+\infty} A_n. Thus \lim_{n\to+\infty} P(B_n)=P(\limsup_{n\to+\infty} A_n)=1. So, for all \epsilon>0, there is M so that 1\geq P(\cup_{i=1}^{+\infty} A_i)\geq P(\cup_{i=M}^{+\infty}A_i)\geq 1-\epsilon$, so P(\cup_{i=1}^{+\infty} A_i)=1.
    I am having trouble with the other direction. Any hints?
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: P[limsup] of independent events
  1. P value (stats) (Replies: 0)

  2. P-adic convergence (Replies: 0)

  3. Weierstrass p function (Replies: 0)

Loading...