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Probability - almost sure convergence

  1. May 11, 2012 #1
    1. The problem statement, all variables and given/known data

    We have ##\mathbb{P}(X_n = 1) = p_n ## and ##P(X_n=0) = 1-p_n ## the question is about almost sure convergence. i.e. does ## X_n \overset{a.s.}{\longrightarrow} 0 ## if ##p_n = 1/n##?

    2. Relevant equations

    ##X_n \overset{a.s.}{\longrightarrow } X ## if ## \mathbb{P}( \omega \in \Omega : X_n(\omega) \to X(\omega) \text{ as } n\to \infty) = 1 ##

    3. The attempt at a solution

    I don't think I understand this properly. Looking at my attempt I've tried a quick ##\epsilon -\delta## setting ##\epsilon = 2/N ## and having ## |1/n| < \epsilon ## for ##n>N##

    I don't think this is what it's asking. Can I say that ## X(\omega) = 0 ## "clearly" and then that ##\mathbb{P}( \omega \in \Omega : |X_n(\omega) - X(\omega)| > \epsilon \text{ i.o. }) = 0## ?

    Where i.o. means infinitely often.
     
  2. jcsd
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