Will a random walk hit every point infinitely often?

[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?

 

[mfb]

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]

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.

 

[mfb]

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]

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?

 

[mfb]

According to wikipedia:

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