  We study the stationary fluctuations of independent run-and-tumble particles.
We prove that the joint densities of particles with given internal state
converges to an infinite dimensional Ornstein-Uhlenbeck process. We also
consider an interacting case, where the particles are subjected to exclusion.
We then study the fluctuations of the total density, which is a non-Markovian
Gaussian process, and obtain its covariance in closed form. By considering
small noise limits of this non-Markovian Gaussian process, we obtain in a
concrete example a large deviation rate function containing memory terms.
