# Limit of passage times of 1-dim random walk

• n0k
In summary, the conversation discusses the problem of finding the limit of a given expression, involving the passage time T_{x} and the expectation E_{0}, as x approaches infinity. The conversation also mentions several attempts at solving the problem, including the use of recurrence relations and functions, but no conclusive solution is reached.
n0k

## Homework Statement

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

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

for any $$\alpha$$ ≥ 0

## 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

What is E0?

E0 is the expectation of hitting zero

## 1. What is a 1-dimensional random walk?

A 1-dimensional random walk is a mathematical model that describes the movement of a particle along a straight line, where each step is determined by a random process.

## 2. What is the limit of passage times in a 1-dimensional random walk?

The limit of passage times in a 1-dimensional random walk refers to the average number of steps it takes for the particle to reach a certain point on the line, as the number of steps approaches infinity.

## 3. How is the limit of passage times calculated in a 1-dimensional random walk?

The limit of passage times is calculated by taking the average of the number of steps it takes for the particle to reach a certain point on the line, over a large number of simulations of the random walk.

## 4. What factors can affect the limit of passage times in a 1-dimensional random walk?

The limit of passage times can be affected by the step size of the particle, the starting position, and the probability distribution of the random process that determines each step.

## 5. What are some real-world applications of studying the limit of passage times in a 1-dimensional random walk?

The limit of passage times in a 1-dimensional random walk has applications in various fields such as physics, biology, and finance. It can be used to model diffusion of particles, movement of molecules in a liquid, and stock prices in financial markets.

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