A Generalization of a Random Walk Problem

Let X _1,X 2,… be i.i.d. random variables with P(X ₁=-k) ∈ (0,1) for some k ∈ N, S ₁=X ₁ +···+ X ₁. We derive an exact expression for the probability that a particle performing a simple random walk will never cross a given straight line, i.e., P(S_l > lα + β for some l ∈ N), where α ∈ [?k,k], β S 0 are rational. Further the exact distribution of sup {S_l /l|≥1} is obtained.