On suprema of Lévy processes with light tails

Let X(t),t≥0,X(0)=0X(t),t≥0,X(0)=0, be a Lévy process with a spectral Lévy measure ρρ. Assuming that ∫−11|x|ρ(dx)<∞ and the right tail of ρρ is light, we show that in the presence of the Brownian component P(sup0≤t≤1X(t)>u)∼P(X(1)>u) as u→∞u→∞, while in the absence of a Brownian component these tails are not always comparable.