Weak convergence of Lookback payoff with correction term

In this article on the Multilevel Monte Carlo method on page 8, http://people.maths.ox.ac.uk/gilesm/files/mcqmc06.pdf, Giles uses a correction term to improve the weak convergence rate of the lookback option payoff $$P = e^{-rT}(S_T - \min_{0\leq t\leq T} S_t)$$ when using the approximation $$\hat{S}_{\min} = \min_{0\leq n \leq N}(\hat{S}_n-\beta^{*}b(nh,\hat{S}_n)\sqrt{h})$$ where $\hat{S}_n$ are the grid values of a discretized diffusion process $S_t$ defined via $dS_t = a(t,S_t)dt+b(t,S_t)$ using the step size $h$ and with $\beta^{*}\approx0.58$. This correction term was introduced by Broadie, Glasserman and Kou, and extended by Gobet in http://hal.archives-ouvertes.fr/docs/00/39/64/22/PDF/BoundaryCorrectionGobetMenozziSPAJune09.pdf.

As proven in Gobet's article and noted in the article by Giles, the correction term improves the discretization error to $o(\sqrt{h})$ from the original order $O(\sqrt{h})$. Giles subsequently concludes that this restores overall $O(h)$ weak convergence from the original weak convergence order $O(\sqrt{h})$. I do not understand how a discretization error of $o(\sqrt{h})$ is sufficient to achieve the overall convergence order. How can removing the leading error term in the discretization improve the result to such a degree?