Visible lattice points in random walks

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 computable) polynomial in alpha of degree 2 left perpendicular (k - 1)/2 right perpendicular. 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. (C) 2018 Elsevier Ltd. All rights reserved.