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