Visible lattice points in random walks

We show that, almost surely, the asymptotic proportion of strings of k consecutive visible lattice points visited by such an α-random walk is a certain constant ck(α), which is actually an (explicitly computable) polynomial in α of degree 2⌊(k−1)∕2⌋. For k=1, this gives that, almost surely, the asymptotic proportion of time the random walker is visible from the origin is c1(α)=6∕π2, independently of α.