Limit theorems for the first passage times of a certain level

Let $\{X_n\}_{n=1,2,\dots,}$ be a sequence of i.i.d.r.v.'s and $T(r)=\min\{k\ge 1:X_1+\dots+X_k\ge r\}$ be the first passage time of a level $r>0$. Two estimates are obtained for the rate of convergence of $T(r)$ to limit laws as $r\to\infty$.