Let $X_1, X_2, \cdots$ be i.i.d. with a finite positive mean $\mu$ and a finite positive variance $\sigma^2$ and let $S_n = X_1 + \cdots + X_n, n \geqq 1$. Further, let $0 \leqq \alpha < 1$ and $t_c$ be the first $n \geqq 1$ for which $S_n > cn^\alpha$ and let $W_c(a) = \sum^\infty_{n = 1} P\{t_c > n, c(n + 1)^\alpha - S_n < a\}$. Under some additional conditions on the distribution of $X_1$ we show that $W_c$ converges weakly to a limit $W$, where $W'(a) = \beta\mu^{-1}P\{S_k \geqq (k + 1)\alpha\mu - a$, for all $k \geqq 0\}$, with $\beta = 1/(1 - \alpha)$. We then find the asymptotic distribution of the excess $R_c = S_{t_c} - ct_c^\alpha$ and show that $R_c$ is asymptotically independent of $t_c^\ast = (t_c - E(t_c))/E(t_c)^{\frac{1}{2}}$, and we compute $E(t_c)$ up to terms which are $o(1)$ as $c \rightarrow \infty$.