Limit of passage times of 1-dim random walk

[n0k]

Homework Statement

T_{x} is the passage time min{n : x(n) = x} for paths starting from x(0)=0

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

for any \alpha ≥ 0

Homework Equations

The Attempt at a Solution

I came up with the recurrence relation of T_{x} as: T_{x} = T_{x-1}+T_{1} = xT_{1}

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

but that doesn't feel right

I also tried f(x) = \frac{e^\alpha}{2}(f(x+1)+f(x-1)) and \frac{T_{x}}{x^2} = \frac{\sum_{i=1}^{x}T_{1}^(i)}{x^2} but again, don't know what to do from there

 

[Mark44]

What is E₀?

 

[n0k]

E₀ is the expectation of hitting zero