# Will a random walk hit every point infinitely often?

• hyurnat4
In summary, the conversation discusses the concept of a random walk with mean 0 and steps that have a positive probability of being greater than a given threshold, and whether such a process will be unbounded and cross the threshold infinitely many times. The conversation also explores the set of allowed step sizes and its distribution, and whether it is possible to hit every point with an infinite number of steps. One participant mentions a well-known result for a step size of 1 and how it can be scaled to any small positive fraction, and the need to show that additional steps do not change the result. The conversation ends with a question about the name of this result, with some suggestions offered.
hyurnat4
Well, not quite a random walk. The steps aren't necessarily ±1, but they have mean 0 and will take values >ε with positive probability. It seems intuitive that such a process will be unbounded and will cross this bound infinitely many times (in 1D). Does anyone know of a result that says this?

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What is the set of allowed step sizes, and what is its distribution? If you consider all real numbers, it is impossible to hit every point as there are uncountable points and countable steps.

hyurnat4
mfb said:
What is the set of allowed step sizes, and what is its distribution? If you consider all real numbers, it is impossible to hit every point as there are uncountable points and countable steps.

Sorry, it was a bad title. I meant 'cross every bound' infinitely often.

For a step size of 1 it is a well-known result, even if just a small positive fraction of all steps is 1 and the rest is zero. This is trivial to scale to ε. All you have to do is to show that additional small steps instead of zero and additional larger steps instead of ε don't change that result.

hyurnat4
mfb said:
For a step size of 1 it is a well-known result, even if just a small positive fraction of all steps is 1 and the rest is zero. This is trivial to scale to ε. All you have to do is to show that additional small steps instead of zero and additional larger steps instead of ε don't change that result.

Ah thank you. Does the result have a name?

According to wikipedia:
This result has many names: the level-crossing phenomenon, recurrence or the gambler's ruin.

## 1. What is a random walk?

A random walk is a mathematical concept that models the path of a particle or object that moves randomly in a given space. It is often used to study the behavior of systems in physics, mathematics, and other fields.

## 2. What does it mean for a random walk to "hit every point infinitely often"?

A random walk hitting every point infinitely often means that, given enough time, the path of the random walk will cover every possible point in the space it is moving in an infinite number of times. Essentially, the random walk will never "get stuck" in one area and will eventually visit every point.

## 3. How is this concept relevant in science?

Random walks have many applications in science, including studying the behavior of molecules, particles, and other physical systems. They can also be used to model the movement of animals, the spread of diseases, and the behavior of financial markets.

## 4. Is it guaranteed that a random walk will hit every point infinitely often?

No, it is not guaranteed. The likelihood of a random walk hitting every point infinitely often depends on various factors, such as the size of the space it is moving in and the rules governing its movement. In some cases, a random walk may get stuck in certain areas and not cover every point infinitely often.

## 5. Are there any real-life examples of random walks hitting every point infinitely often?

Yes, there are many real-life examples of random walks hitting every point infinitely often. One example is the movement of molecules in a gas, which can be modeled as a random walk and will eventually cover every point in the container. Another example is the behavior of stock market prices, which can be modeled as a random walk and will eventually hit every possible price infinitely often.

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