Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 209 / 1 / b / Solution

Use the same $M_n,v_i,S,\Delta$ as in part (a). The error of the weighted realized variance splits into its centered fluctuation and its deterministic quadrature error:The Hölder continuity of $g$ controls this bias:Since $\mathbb{E}{M}_n=0$, the bias-variance decomposition of mean squared error has no cross term. Part (a) therefore givesHence $\tilde{D}=3$ is a universal choice. This argument separately controls the statistical fluctuation and the Riemann sum approximation.