The overshoot of a random walk with negative drift

Let {Sn,n⩾0}{Sn,n⩾0} be a random walk starting from 0 and drifting to -∞-∞, and let τ(x)τ(x) be the first time when the random walk crosses a given level x⩾0x⩾0. Some asymptotics for the tail probability of the overshoot Sτ(x)-xSτ(x)-x, associated with the event (τ(x)<∞)(τ(x)<∞), are derived for the cases of heavy-tailed and light-tailed increments. In particular, the formulae obtained fulfill certain uniform requirements.