SCHWEITZER 2006 Problem 11

Let $\alpha$ be an irrational number, and denote $F = \{ (x,y) \in R^2 : y \geq \alpha x \}$ as a closed half-plane bounded by a line. Let $P(\alpha,n) = P(X_1,...,X_n \in F)$ , where $X_n$ is a simple, symmetric random walk that starts at the origin and moves with probability 1/4 in each direction. Prove that $P(\alpha,n)$ does not depend on $\alpha$ .