Limit of passage times of 1-dim random walk

  • Thread starter n0k
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  • #1
n0k
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Homework Statement


[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


Homework Equations





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
 

Answers and Replies

  • #3
n0k
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E0 is the expectation of hitting zero
 

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