  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.
