  Associated to a random walk on $\mathbb{Z}$ and a positive integer $n$, there
is a return probability of the random walk returning to the origin after $n$
steps. An interesting question is when the set of return probabilities uniquely
determines the random walk. There are trivial situations where two different
random walks have the same return probabilities. However, among random walks in
which these situations do not occur, our main result states that the return
probabilities do determine the random walk. As a corollary, we will obtain a
result dealing with the representation theory of $\mathrm{U}(1)$.
