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In this article on the Multilevel Monte Carlo method on page 8,, 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

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?

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