The discussion centers around the behavior of a random walk, particularly whether it will hit every point infinitely often or cross every bound infinitely often. The focus includes theoretical aspects of random walks, step sizes, and their distributions.
Participants express differing views on the implications of step sizes and distributions, with no consensus reached on whether every point can be hit or only bounds crossed infinitely often.
There are limitations regarding the definitions of step sizes and their distributions, as well as the implications of countability versus uncountability in the context of random walks.
This discussion may be of interest to those studying stochastic processes, random walks, or mathematical theories related to probability and recurrence.
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.