Consider a sequence of confidence intervals {In}. Sequential procedures are based on stopping rules which, for specified constantswandp, terminate sampling at a valuenfor whichInhas width approximatelywand noncoverage probability approximatelyp. Asymptotic analyses of such procedures have previously concerned onlyw→0withpfixed. This article presents a novel and general approach for the casep→ 0 withwfixed. Application is made to procedures based on the sample mean and median, and the corresponding asymptotic relative efficiencies are evaluated for selected values ofwfor the normal, Laplace, logistic, and uniform distributions.