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Limit of passage times of 1-dim random walk

  1. Dec 11, 2009 #1

    n0k

    User Avatar

    1. The problem statement, all variables and given/known data
    [tex]T_{x}[/tex] is the passage time min{n : x(n) = x} for paths starting from x(0)=0

    Show:
    [tex]
    \lim_{x\rightarrow +\infty}E_{0}(e^\frac{-\alpha\\T_{x}}{x^2}) = e^\-\sqrt{2\alpha}
    [/tex]

    for any [tex]\alpha[/tex] ≥ 0


    2. Relevant equations



    3. The attempt at a solution
    I came up with the recurrence relation of T_{x} as: [tex]T_{x} = T_{x-1}+T_{1} = xT_{1}[/tex]

    so [tex]E_{0}(e^\frac{-\alpha\\T_{x}}{x^2})[/tex] becomes [tex]E_{0}(e^\frac{-\alpha\\T_{1}}{x})[/tex]

    but that doesn't feel right

    I also tried [tex]f(x) = \frac{e^\alpha}{2}(f(x+1)+f(x-1))[/tex] and [tex]\frac{T_{x}}{x^2} = \frac{\sum_{i=1}^{x}T_{1}^(i)}{x^2}[/tex] but again, don't know what to do from there
     
  2. jcsd
  3. Dec 12, 2009 #2

    Mark44

    Staff: Mentor

    What is E0?
     
  4. Dec 12, 2009 #3

    n0k

    User Avatar

    E0 is the expectation of hitting zero
     
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