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Independant RV's and the Borel-Cantelli Lemmas

  1. Jan 15, 2012 #1
    wvcuvp.png

    The first part I can do no problem!

    The second part... well I've really no idea what to do with that S term, I assume it has something to do with the strong law of large numbers? (w is an element in the set of outcomes)

    Borel-Cantelli Lemmas

    Law of Large Numbers

    Does anyone know where to start?

    Thanks

    edit:

    I have an exmple of strong convergence showing that

    if P(w : lim Xn(w) = X(w) )

    then this is identically equal to

    P(Xn -> X) = 1 , almost surely

    So I guess I just have to show that P(Xn -> X) = 1,

    So show: P(Sn(w)/n -> -1) = 1?

    Probably going the wrong way around it.. not sure how Borel Cantelli links to it.
     
    Last edited: Jan 15, 2012
  2. jcsd
  3. Jan 15, 2012 #2

    Ray Vickson

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    It is probably simpler to re-write the problem a bit: let [itex]X_n = -1 + Y_n,[/itex] where [tex] \Pr\{Y_n = 0 \} = 1 - \frac{1}{n^2}, \mbox{ and } \Pr \{Y_n = n^2 \} = \frac{1}{n^2}. [/tex] Essentially, you want to show that [itex] \sum_{k=1}^n Y_k / n \rightarrow 0 [/itex] w.p. 1. If [itex] A = \{ Y_n > 0 \mbox{ infinitely often } \},[/itex] can you show that Pr{A} = 0?

    RGV
     
  4. Jan 15, 2012 #3
    Hi RGV, you always answer my questions it seems! :)

    So I need to show given the event En = Yn>0

    that the sum from n=1 to inf of P(En) is finite (Borel Cantelli)

    So this is the sum of n=1 to inf of 1/n2?

    Which is pi^2/6, and so the probability is 0!

    Correct?
     
  5. Jan 15, 2012 #4
    Also any idea on the second part of this question?

    v5ltep.png

    It looks like the same structure..

    Thanks!
     
  6. Jan 16, 2012 #5
    bump..!
     
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