Every $2\times2$ principal <submatrix> of a <kernel matrix> is positive semidefinite, so
$$
|k_\tau(x,y)|^2\leq k_\tau(x,x)k_\tau(y,y).
$$
The <Cauchy-Schwarz inequality> for integrals therefore gives
$$
\int_{-\infty}^{\infty}|k_\tau(x,y)|\,d\tau
\leq
\left(\int_{-\infty}^{\infty}k_\tau(x,x)\,d\tau\right)^{1/2}
\left(\int_{-\infty}^{\infty}k_\tau(y,y)\,d\tau\right)^{1/2}<\infty.
$$
Thus every entry of $k(x,y)=\int k_\tau(x,y)d\tau$ is well defined. For any finite coefficients $c_r$ and points $x_r$, linearity of the integral gives
$$
\sum_{r,s}c_rc_s k(x_r,x_s)
=\int_{-\infty}^{\infty}\sum_{r,s}c_rc_s k_\tau(x_r,x_s)\,d\tau\geq0,
$$
because the integrand is nonnegative. Hence $k$ is a <positive-semidefinite kernel>. This proves the <integral closure of positive-semidefinite kernels>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 205 / 1 / a / ii / Solution

{wiki}

A submatrix is obtained from a <matrix> by retaining selected rows and selected columns. A principal submatrix of a square matrix retains the same index set for its rows and columns.

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