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Math Help - Probability of Eventual Return in Random Walk

  1. #1
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    Probability of Eventual Return in Random Walk

    The probability of eventual return is given by:
    f_{2} + f_{4} + f_{6} + ...+ f_{\infty} = 1

    where f_{n} = Probability of return at Time Period n.
    Note that returns must be in even numbers, as a +1 needs -1 to cancel each other out.

    Ok, I understand the theory, but I cannot find a suitable proof.


    Edit: Proof found. Problem solved.

    f_{n} is given by f_{2n}= u_{2n-2} - u_{2n} where u_{2n}={2n \choose n}2^{-2n}

    So, f_{2} + f_{4} + f_{6} + ...+ f_{\infty} = u_{0} - u_{2} + u_{2} - u_{4} + ..... = u_{0} = 1


    Still, I'll be very grateful if anyone can explain this Theorem:
    f_{2n}= u_{2n-2} - u_{2n}
    Last edited by chopet; November 21st 2007 at 11:43 AM.
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