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Math Help - random signs

  1. #1
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    random signs

    Let (X_n)_{n \geq 0} be i.i.d random signs, P(X_n=1)=P(X_{n-1}=-1)=1/2. Let (a_n)_{n \geq 0} be a sequence of real numbers such that \Sigma _{n=1} ^{\infty}a_n ^2 < \infty. Show that \Sigma _{n=1} ^ \infty a_n X_n is almost sure convergent.

    By \Sigma _{n=1} ^{\infty}a_n ^2 < \infty, a_n \rightarrow 0 as n \rightarrow \infty. but i dont know how to go from here.
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  2. #2
    MHF Contributor matheagle's Avatar
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    Have you seen the Khintchine-Kolmogorov Convergence Theorem?
    It follows directly from that.
    I found it online RIGHT out of Chow-Teicher.
    http://books.google.com/books?id=mYn...heorem&f=false
    I have all three editions of that book.
    My copy from 78 is falling apart.
    Teicher is my advisor's advisor and I published about 15 times In Chow's journal in Taiwan.
    http://en.wikipedia.org/wiki/Yuan-Shih_Chow
    (Academia Sinica)
    So, I use their book a lot.
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