Suppose I have a discrete random walk with equal probabilities, and the particle begins at $x=0$. The process goes on indefinitely. What is the probability that the particle will never leave $[-n,n]$, where $n \in\mathbb{N}$ ?
2026-04-03 19:46:03.1775245563
Discrete random walk bounded in an interval
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You will stay in that interval in infinitely many steps with probability $0$. In fact, with probability $1$ the walk visits every point infinitely many times.
Proving this is a bit long for this site, but there's a survey paper that discusses random walks here.