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Lim sup and lim inf of IID RVs

  1. May 5, 2009 #1
    [tex]X_1, X_2, \ldots[/tex] are iid random variables with [tex]P ( X_1 = n ) = P ( X_1 = - n ) = \frac{ c }{ n^2 \log n }[/tex] where c makes the probabilities sum to one. Define [tex]S_n = X_1 + \ldots + X_n[/tex]. We want to show that
    [tex]\limsup \frac{S_n}{n} =\infty[/tex] and [tex]\liminf \frac{S_n}{n} = -\infty[/tex] almost surely.

    I've managed to use the Borel-Cantelli lemma to show that [tex]P(|X_n| \geq n \text{ infinitely often}) = 1[/tex], but I can't pass to the lim sup/inf. Any help/suggestions would be appreciated.
  2. jcsd
  3. May 18, 2009 #2
    Suppose limsup S(n)/n is not infinity; instead limsup S(n)/n = m where m is finite...
  4. May 19, 2009 #3


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    Note that if S(n)/n were bounded then X(n)/n = S(n) - (1-1/n)S(n-1) would also be bounded. Can you show that this is false?
    Once that's done, Kolmogorov's 0-1 law should finish it off.
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