For any $x_1,\ldots,x_N$ and $a_1,\ldots,a_N$, part (b) and linearity of expectation give
$$
\sum_{i,j}a_ia_jk(x_i,x_j)
=\mathbb E\left[\left(\sum_i a_i\cos(Wx_i+U)\right)^2\right]\geq0.
$$
Symmetry is immediate, so $k$ is a positive-definite kernel.


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 205 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-205/3/c/solution