Several stopping times which arise from problems of sequential estimation may be written in the form $t_c = \inf\{n \geqq m: S_n < cn^\alpha L(n)\}$ where $S_n, n \geqq 1,$ are the partial sums of i.i.d. positive random variables, $\alpha > 1, L(n)$ is a convergent sequence, and $c$ is a positive parameter which is often allowed to approach zero. In this paper we find the asymptotic distribution of the excess $R_c = ct_c^\alpha - S_{t_c}$ as $c \rightarrow 0$ and use it to obtain sharp estimates for $E\{t_c\}.$ We then apply our results to obtain second order approximations to the expected sample size and risk of some sequential procedures for estimation.