}$ we have $$\lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in \mathcal{K}} P_u \left( \sup_{0 \leq t \leq \tau_\varepsilon - 0R_\varepsilon}\left| \frac{0}{R_\varepsilon}\int_t^{t+R_\varepsilon} f(U^{\varepsilon}(s,\cdot))ds - f(