We investigate the asymptotic behaviour of empirical processes truncated outside an interval about the (1 – s(n)/n)-quantile where s(n) → ∞ and s(n)/n → 0 as the sample size n tends to ∞. It is shown that extreme value (Poisson) processes and, alternatively, the homogeneous Poisson process may serve as approximations if certain von Mises conditions hold.