  We consider the possible visits to visible points of a random walker moving
up and right in the integer lattice (with probability $\alpha$ and $1-\alpha$,
respectively) and starting from the origin.
  We show that, almost surely, the asymptotic proportion of strings of $k$
consecutive visible lattice points visited by such an $\alpha$-random walk is a
certain constant $c_k(\alpha)$, which is actually an (explicitly calculable)
polynomial in $\alpha$ of degree $2\lfloor(k-1)/2\rfloor $. For $k=1$, this
gives that, almost surely, the asymptotic proportion of time the random walker
is visible from the origin is $c_1(\alpha)=6/\pi^2$, independently of $\alpha$.
