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Regard each centered random variable $g(x)$ as a vector $h_x$ in the Hilbert space $L^2(\mathbb{P})$. Thenso $k$ is the Gaussian kernel on the finite subset $\{h_x:x\in \mathcal{X}\}$ of that Hilbert space. More explicitly,Every power of the inner-product kernel is positive semidefinite, and the closure property of positive-semidefinite kernels under nonnegative sums, pointwise limits, and multiplication by one-variable factors proves that $k$ is positive definite.

Let $V_1,\dots ,V_d$ be independent standard Cauchy random variables. Their characteristic functions and independence giveTaking real parts yieldsFor each realization of $V$, both products are rank-one positive-semidefinite kernels. Their sum and then their expectation remain positive semidefinite by the closure property of positive-semidefinite kernels. This is a Random Fourier feature representation of the Laplace kernel.