This is just some question that popped out of nowhere while starting studying random walks, and I don't really know how to approach this.
Say I have a random walk that starts at zero, and goes up or down by one at each step with equal probability.
For some integer $i$, we stop the walk whenever it reaches either $i$ or $-i$.
Suppose we are given that the walk stopped by reaching $i$. I'm interested in the minimum value the walk passed through. In other words, for some $0 \geq j > -i$, what it the probability that the walk took value $j$ at some point, but not $j - 1$ ?