Say a 1D random walk starts at the origin. At every time step, the random walk advances $+1$ with probability $\alpha$, stays at its current location with probability $\beta$, and returns to the origin with probability $1-\alpha-\beta$. Once the random walk returns to the origin, the walk is over, no future movement occurs.
My question is, what is the distribution of the maximum distance from the origin reached on the random walk? Is it perhaps an exponential or power-law distribution?
Thank you for any suggestions or feedback.