The <Lipschitz continuity> of $p_1$ gives
$$
|e_1(x)-p_1(x)|
\leq\frac CL\sum_{\ell=1}^L
\lVert X_{(\ell)}-x\rVert_\infty
\leq C\lVert X_{(L)}-x\rVert_\infty.
$$
Part (d) therefore implies
$$
\mathbb E|e_1(x)-p_1(x)|
\leq C\left\{
\left(\frac{2L}{nM}\right)^{1/d}+\frac2L
\right\}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 2 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-218/2/e/solution