Gaussian convergence for stochastic acceleration of $\cN$ particles in the dense spectrum limit

The velocity of a passive particle in a one-dimensional wave field is shown to converge in law to a Wiener process, in the limit of a dense wave spectrum with independent complex amplitudes, where the random phases distribution is invariant modulo $\pi/2$ and the power spectrum expectation is uniform. The proof provides a full probabilistic foundation to the quasilinear approximation in this limit. The result extends to an arbitrary number of particles, founding the use of the ensemble picture for their behaviour in a single realization of the stochastic wave field.