Before revision:  Moreover, we show that the rate of convergence for the ruin probability is of order O(n^{-1/2 \mO\big(}\big) , and we show that the convergence is {\it uniform} with respect to the surplus.
After revision:  Moreover, we show that the rate of convergence for the ruin probability is of order O(n^{-1/2 \mO\big(n^{-1/2}\big) , and we show that the convergence is {\it uniform} with respect to the surplus.