Estimates for the overshoot of a random walk with negative drift and non-convolution equivalent increments

Let {Sn:n≥0} be a random walk with negative drift and τ(x) be the first time when the random walk crosses a given level x≥0. This paper focuses on random walks with non-convolution equivalent increments. For this random walk, the uniform asymptotics of P(Sτ(x)−x>y,τ(x)<∞), as x→∞, have been presented.