The first passage time problem over a moving boundary for asymptotically stable Lévy processes

Exact asymptotics for the instant of crossing a curve boundary by an asymptotically stable random walk

“…However, recent results by Aurzada and Kramm [2] give strong grounds to suspect that conditions in [9] are not optimal. More precisely, it is shown in [2] that if g(t) = o(t γ ) with some γ < 1/α then P(T g > n) and P(T 0 > n) have the same rough asymptotics, that is,…”