# Range of random walk

I have a random walk on $\mathbb{Z}$ with starting point $0$ and with length $n$ and possible steps to right, left or stay where you are, all with the same probabilities. I am interested in exact probability that the walk will visit $k$ distinct values.

I think this should be known but I cannot find such result.

• Not an answer, but you should check out the reflexion principle for Brownian motion (obviously the analogous result for a simple random walk is also true) since it allows one to calculate the exact distribution of the maximum value of the random walk on an interval (and the minimum, but in principle you would need the joint distribution of both to answer your question). This should give you an idea of what to expect for large $n$ and $k$ of order $\sqrt{n}$. May 23, 2016 at 11:57