We present a class of space-time processes, which can be viewed as functional autoregressions taking values in the space of square integrable functions on the sphere. We exploit some natural isotropy requirements to obtain a neat expression for the autoregressive functionals, which are then estimated by a form of frequency domain least squares. For our estimators, we are able to show consistency and limiting distributions. We prove indeed a quantitative version of the central limit theorem, thus deriving explicit bounds (in Wasserstein metric) for the rate of convergence to the limiting Gaussian distribution; to this aim we exploit the rich machinery of Stein-Malliavin methods. Our results are then illustrated by numerical simulations
