Gaussian coordinates of deterministic orthonormal Wiener integrands (${\xi }_n=\int g_{n}dW\text{independent }N(0,1)$)

Deterministic Itô integrals are jointly centered Gaussian by approximation from step functions and the Itô isometry. Their covariance is the L2 inner product of their integrands. Orthonormal integrands therefore produce independent standard-normal coordinates, with independence of a countable family understood through all finite subfamilies.