On convergence of some normalized extreme order statistics

We investigate normalized extreme order statistics for distributions supported on a finite interval. Under local endpoint assumptions of the form $f(x)\sim C(b-x{)}^{r-1}(x\to b-),$ and its left-endpoint analogue, we prove that the normalized sample maximum and minimum are uniformly integrable in every positive power. As a consequence, weak convergence upgrades to convergence of all positive moments. We then establish the joint weak convergence of the normalized lower and upper extremes, and prove that they are asymptotically independent. This leads to the limit law of their difference and an explicit asymptotic variance formula. These results are applied to the sample midrange in symmetric bounded location families. In particular, when the endpoint exponent satisfies $1\le r<2$, the sample midrange has asymptotic variance of order $n^{-2/r}$ and is therefore asymptotically more efficient than the sample mean and interior sample quantiles, whose variances are typically of order $n^{-1}$. We also obtain the asymptotic law of the sample range as a natural companion result. The main statistical application remains the sample midrange as an estimator of the center parameter. Semicircular and geometric localization examples are discussed, and a simulation study confirms the asymptotic results.