Let X1,X2, ... be independent, identically distributed with finite mean μ > 0, Sn = X1 +···+ Xn. For f(n) = nβ, c > 0 we consider the stopping times Tc = inf {n : Sn > c + f(n)} with overshoot Rc = STc – (c + f(Tc)). For 0 < β < 1 we give a bound for supc ≥ 0 ERc in the spirit of Lorden's well-known inequality for f = 0.
